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554 I Sc ienti fic Library % — I OMIIEB STATES PATENT OFFICE 11 — 8620 government frintixq office THE LONDON, EDINBURGH, and DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. CONDUCTED BY SIR OLIVER JOSEPH LODGE, D.Sc, LL.D., F.R.B. SIR JOSEPH JOHN THOMSON, O.M., M.A., So.D., LL.D., F.R.S. JOHN JOLY, M.A., D.Sc, F.R.S., F.G.S. RICHARD TAUNTON FRANCIS AND WILLIAM FRANCIS, F.L.S. "Nee araisearuni sane textus ideo melior quia ex se ftla gigiiunt, nee noster ilior quia ^x alienis libamus ut apes." Just. Lips. Polit. lib. L cap. 1. Not. VOL. XLIV.— SIXTH SERIES. JULY— DECEMBER 1922. LONDON-. TAYLOR AND FRANCIS, RED LION COURT. FLEET STREET. SOLD BY SMITH AND SON, GLASGOW ;— HODGES, FIGGIS, AND CO., DUBLIN ; - AND VEUVE J. BOYVEAU, PAKI C , "Ateditatiuins est perscrutari occulta ; contemplationis est admirari perspicua '. . . . Admiratio generat qiuestionem, quzestio investigationero, investigalio inventionem." — Hugo de S. Victore. — — " Cur spirent venti, cur terra dehiscat, Our in are turgescat, pelago cur tantus amaror, Cur caput obscura Phoebus ferrugine condat, Quid toties diros cogat -flag-rare cometas, Quid pariat nubes, veniant cur fulmina coelo, Quo micet igne Iris, superos quis conciat orbes Tain vario motu." J, B. Pinelli ad Muzonium. ATiERK CONTENTS OF VOL. XLIV (SIXTH SERIES). '■221. NUMBER CCLIX.— JULY 1922. Mr. A. R. McLeod on the Unsteady Motion produced in a Uniformly Rotating Cylinder of Water by a Sudden Change in the Angular Velocity of the Boundary 1 Dr. Norman Campbell on the Elements of Geometry 15 Dr. Dorothy Written on the Rotation of Slightly Elastic Bodies ... 30 Mr. G. H. Henderson on the Straggling of ex. Particles by Matter . . 42 Dr. G. Green on Fluid Motion relative to a Rotating Earth 52 Prof. G. N. Antonoffon the Breaking Stress of Crystals of Rock-Salt. 62 Dr. Norman Campbell on the Measurement of Chance 07 Messrs. H. T. Tizard and D. R. Pye : Experiments on the Ignition of Gases by Sudden Compression. (Plate I ) 79 Mr. C. Rodgers on the Vibration and Critical Speeds of Rotors. . . . 122 Mr. P. Cormack on Harmonic Analysis of Motion transmitted by Hooke's Joint 156 Messrs. E. W. B. Gill and J. H. Morrell on Short Electric Waves obtained by Valves 161 Sir George Greenhill on Pseudo-Regular Precession 179 Dr. J. \Y. Nicholson on the Binding of Atoms by Electrons 193 Mr. L. St. C. Broughall on Theoretical Aspects of the Neon Spectrum. 2U4 Dr. F. H. Newman on Absorption of Hydrogen by Elements in the Electric Discharge Tube 215 Mr. B. A. M. Cavanagh on Molecular Thermodynamics. II 226 Prof. J. G. Gray on the Calculation of Centroids 247 Messrs. A. P. II. Trivelli and L. Righter : Preliminary Investiga- tions on Silberstein's Quantum Theory of Photographic Exposure. 252 Dr. L. Silberstein on a Quantum Theory of Photographic Exposure. 257 Mr. R. F. Gwyther on an Analytical Discrimination of Elastic Stresses in an Isotropic Body 274 Mr. II. S. Rowell on Damped Vibrations 284 Notices respecting New Books : — Mr. L. F. Richardson's Weather Prediction by Numerical Process 285 Proceedings of the Geological Society : — Mr. C. E. No will Bromehead on the Influence of Geology on the History of London 286 Intelligence and Miscellaneous Articles-. — On Young's Modulus and Poisson's Ratio for Spruce, by Mr. II. Carrington 288 IV CONTENTS OF VOL. XL1V. SIXTH SERIES. NUMBER OOLX.— AUGUST. Page Mr. C. J. Smith on the Viscosity and Molecular Dimensions of Gaseous Carbon Oxy sulphide (COS) 289 Prof. A. 0. Rankine on the Molecular Structure of Uarbon Oxy- sulphide and Carbon Bisulphide 292 Mr. F. P. Slater on the Rise of y Ray Activity of Radium Emanation. 300 Profs. J. N. Mukherjee and B. C. Papaconstantinou on an Experi- mental Test of Smoluchowski's Theory of the Kinetics of the Process of Coagulation 305 Prof. J. N. Mukherjee on the Adsorption of Ions 32L Prof. W. M. Hicks on certain Assumptions in the Quantum-Orbit Theory of Spectra 346 Mr. F. C. Toy on the Theory of the Characteristic Curve of a Photo- graphic Emulsion 352 Mr. A. M. Mosharrafa on the Stark Effect for Strong Electric Fields ■ , . 371 Mr. E. Takagishi on the Damping Coefficients of the Oscillations in Three-Coupled Electric Circuits 373 Prof. S. C. Kar on the Electrodjnamic Potentials of Moving Charges ■ . . . 376 Mr. A. E. Harward on the Identical Relations in Einstein's Theory. 380 Mr. H. S. Rowell un Energy Partition in the Double Pendulum . . . 382 Piof. J. S. Townsend on the Velocity of Elections in Gases 384 Prof. H. A. McTaggart on the Electrification at the Boundary between a Liquid and a Gas 386 Prof. L. V. King on a Lecture-Room Demonstration of Atomic Models. .Plate II.) 395 Mr. 11. D. Murray on the Influence of the Size of Colloid Particles upon the Adsorption of Electrolytes 401 Notices respecting New Books: — Dr. G. Scott Robertson's Basic Slags and Rock Phosphates . . . 415 NUMBER CCLXI.— SEPTEMBER, Prcf. Sir E. Rutherford and Dr. J. Chadwick on the Disintegration of Elements by oc Particles 417 Prof. W. L. Bragg and Messrs. R. W. James and 0. II. Bosanquet on the Distribution of Electrons around the Nucleus in the Sodium and Chlorine Atoms 433 Messrs. C. G. Darwin and R. H. Fowler on the Partition of Energy 450 Mr. M. II. Belz on the Heterodyne Beat Method and some Appli- cations to Physical Measurements 479 Mr. R. F. Gwyther on the Conditions for Elastic Equilibrium under Surface Tractions in a Uniformly Eolotropic Body 501 Mr. C. J. Smith on the Viscosity and Molecular Dimensions of Sulphur Dioxide 508 Mr. S. Lees on a Simple Model to Illustrate Elaslic Hysteresis. . . . 511 Prof. R. W. Wood on Atomic Hydrogen and the Buhner Series Spectrum 538 Mr. D. Coster on the Spectra of X-rays and the Theory of Atomic • Structure 546 Pro£ E. H. Barton and Dr. H. M. Browning on Vibrational Re- ' sponders under Compound Forcing. (Plates III. & IV.) ...... 573 CONTENTS OF VOL. XLIV. SIXTH SERIES. V Page Dr. Norman Campbell and Mr. 13. P. Dudding on the Measurement of Light 577 Mr. D. L. llammick on Latent Heats of Vaporization and Expansion 590 The late William Gordon Brown on the Faraday-Tube Theory of Electro-Magnetism , . 594 Mr. B. A. M. Cavanagh ou Molecular Thermodynamics. Ill 610 Prof. A. W. Porter and Mr. J. J. Hedges on the Law of Distribution of Particles in Colloidal Suspensions, with Special Reference to Perrin's Investigations , (541 Mr. St. Landau-Ziemecki on the Emission Spectrum of Monatomic Iodine Vapour 651 NUMBER COLXII.— OCTOBER. Sir J. J. Tuomson : Further Studies on the Electron Theory of Solids, The Compressibilities ot a Divalent Metal and of the Diamond. Electric and Thermal Conductivities of Metals 657 Dr. G. H. Henderson on the Decrease of Energy of ol Particles on passing through Matter 680 Mr. D. C. Henry on a Kinetic Theory of Adsorption 689 Prof. S. R. Milner on Electromagnetic Lines and Tubes 705 Mr. A. Bramley on Radiation 720 Mr. G. Breit on the Effective Capacity of a Pancake Coil 729 Prof. F. E. TIackett on the Relativity-Contraction in a Rotating Shaft moving with Uniform Speed along its Axis 740 Dr. T. J. Baker on Breath Figures 752 -Mr. A. Sellerio on the Repulsive Effect upon the Poles of the Electric Arc 765 Mr. B. B. Baker on the Path of an Electron in the Neighbourhood of an Atom 777 Prof. A. W. Porter and Mr. R. E. Gibbs on the Theory of Freezing Mixtures , . 787 NUMBER CCLXIII.— NOVEMBER. Mr. G. Shearer on the Emission of Elections by X-Rays 793 Mr. A. J. Saxton on Impact Ionization by Low-Speed Positive H-Ions in Hydrogen , 809 Messrs. C. G. Darwin and R. i\. Fowler on the Partition of Energy. — Part II. Statistical Principles and Thermodynamics 823 Mr. J. H. Van Vleck on the normal Helium Atom and its relation to the Quantum Theory 842 Mr. G. A. Tomlinson on the Use of a Triode Valve in registering Electrical Contacts 870 .Mr. E. A. Milne on Radiative Equilibrium : the Insolation of an Atmosphere 872 Dr. S. C. Bradford on the Molecular Theory of Solution. II 897 Mr. R. A. Mallet on the Failure of the Reciprocitv Law in Photography 904 Me-srs. R. W. Rolens, J. II. Smith, and S. S. Richardson on Magnetic Rotatory Dispei s'on of certain Paramagnetic Solutions. 912 VI CONTENTS OF VOL. XLIV. SIXTH SERIES. Page Dr. F. W. Edridge-Green on Colour- Vision Theories in Relation to Colour-Blindness 916 Mr. A. H. Davis on Natural Convective Cooling in Fluids 920 Mr. A. II. Davis on the Cooling Power of a Stream of Viscous Fluid : 940 Dr. F. H. Newman on a Sodium-Potassium Vapour Arc Lamp. (Plate V.) 944 Mr. J. J. Mauley on the Protection of Brass Weights 948 Mr. H. S. Rowell on the Analysis of Damped Vibrations 951 Mr. F. M. Lidstone on the Full Effect of the Variable Head in Viscosity Determinations , 953 Dr. L. Silberstein and Mr. A. P. II. Trivelli on a Quantum Theory of Photographic Exposure. (Second Paper.) ... 956 Dr. J. S. G. Thomas on the Discharge of Air through Small Orifices, and the Entrainment of Air by the .Issuing Jet. (Plate VI.). . . 969 Dr. J. R. Partington on the Chemical Constants of some Diatomic Gases 988 Mr. M. F. Skinner on the Motion of Electrons in Carbon Dioxide. . 994 Mr. W. N. Bond on a Wide Angle Lens for Cloud Recording. (Plate VII.) . . . 999 The Research Staff of the General Electric Company Ltd., London : A Problem in Viscosity : The thickne-s of liquid films formed on solid surfaces under dynamic conditions. (Plate VIII.) 1002 Prof. S. Timoshenko on the Distribution of Stresses in a Circular Ring compressed by Two Forces acting along a Diameter .... 1014 Prof. A. VV. Porter on a Revised Equation of State 1020 Prof. V. Ka rape toff : General liquations of a Balanced Alternating- Current Bridge 1024 Prof. J. S. Towtjsend and Mr. V. A. Bailey on the Motion of Elec- trons in Argon and in Hydrogen 1033 Prof. S. R. Milner : Does an Accelerated Electron necessarily radiate Energy on the Classical Theory ? .\ ..... . 1052. Mr. A. Press on a Simple Model to illustrate Elastic Hysteresis . . 1053 Mr. S, Lees : Note on the above = 1054 Notices respecting New Books : — M. H. Ollivier's Cours de physique generale 1055 Dr. J. W. Mellor's A Comprehensive Treatise on Inorganic and Theoretical Chemistry - 1056 Mr. S. G. Starling's Science in the Service of Man : Electricity. 1058 M. Gustave Mie's La theorie Einsteinienne de la Gravitation. . J 058 Wave-lengths in the Arc Spectra of Yttrium, Lanthanum, and Cerium and the preparation of pure Rare Earth Elements. . 1058 The Journal of Scientific Instruments 1059 Proceedings of the Geological Society : — Mr. R. D. Oldham on the Cause and Character of Earthquakes. 1060 Sir C. J. Holmes on Leonardo da Vinci as a Geologist 1061 Intelligence and Miscellaneous Articles: — On the Buckling of Deep Beams, by Dr. J. Prescott 1062 On Damped Vibrations, by Mr. C. E. Wright 1063 On the Magnetic Properties of the Hydrogen - Palladium System, by Mr. A. E. Oxley 1063 On Short Electric Waves obtained by Valves, by Prof. R. Whiddington 1064 CONTENTS OF VOL. XLIV.— SIXTH SERIES. VII NUMBER CCLXIV,— DECEMBER. Page Mr. 1!. Hargreaves on Atomic Systems based on Free Electrons, positive and negative, and their Stability 1065 Prof. R. AY. Wood on Selective Reflexion of X. 2536 by Mercury Vapour 1105 Prof. R. W. Wood on Polarized Resonance Radiation of Mercury Vapour. (Plate IX.) . ._ _ 1107 Mr. S. J. Barnett on Electric Fields due to the Motion of Constant Electromagnetic Systems 1112 Prof. Megh Nad Saba on the Temperature Ionization of Elements of the Higher Groups in the Periodic Classification 1128 Prof. F. Uorton and Dr. A. C. Davies on the Ionization of Ab- normal Helium Atoms by Low-Voltage Electronic Bombardment. 1140 Prof. J. S. Townsend on the Ionizing Potential of Positive Ions . . 1147 Dr. G. Breit on the Propagation of a Fan-shaped Group of Waves in a Dispersing Medium 1149 Prof. E. Keighlley Rideal on the Flow of Liquids under Capillary Pressure '.1152 Prof. S. Russ and Mr. L. H. Clark on a Balance Method of mea- suring X-Rays 1 1 59 Mr. J. W. T. Walsh on the Measurement of Light 1165 Notices respecting New Books : — The Cambridge Colloquium, 1916 : Part 1 1169 Mr. J. Edwards's A Treatise on the Integral Calculus : Vol. II. 1169 Index 1170 PLATES. I. Illustrative of Messrs. H. T. lizard and D. R. Pye's Paper 0:1 Experiments on the Ignition of Gases by Sudden Com- pression. II. Illustrative of Prof. L. V. King's Paper on a Lecture-Room Demonstration of Atomic Models. III. & IV. Illustrative of Prof. Barton and Dr. B:owning's Paper on Vibrational Responders under Compound Forcing. V. Illustrative of Dr. F. H. Newman's Paper on a Sodium- Potassium Vapour Arc Lamp. VI. Illustrative of Dr. J. S. G. Thomas's Paper on the Discharge of Air through Small Orifices, and the Entrainment of Air by the Issuing Jet. VII. Illustrative of Kir. W. N. Bond's Paper on a Wide Angle Lens for Cloud Recording. VIII. Illustrative of The Research Staff of the General Electric Company on a Problem in Viscosity. IX. Illustrative of Prof. R. W. Wood's Paper on Polarized Reso- nance Radiation of Mercury Vapour, THE LONDON, EDINBURGH, and DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. I. The Unsteady Motion 'produced in a Uniformly Rotating Cylinder of Water by a Sudden Change in the Angular Velocity of the Boundary. By A. R. McLeod, M.A., Fellow of Gonville and Caius College , Cambridge *. IN the following paper a comparison is made between the observed angular velocities in a rotating circular cylinder of water, and those calculated by two-dimensional theory in which the effects of the base and the free surface are neglected, and each particle of water is assumed to move in a circle about the axis of rotation. The two cases dealt with are those of unsteady motion in which the cylindrical containing-wall is suddenly started from rest, or suddenly stopped when, with the water, it is rotating uniformly. The measurements are all of surface velocities, because the use of lycopodium particles floating on the surface was necessary, as liquid globules of the same density as the water would not remain at the same depth for any length of time. Three cylinders of approximate diameters 5 cm., 15 cm., and 25 cm. were used, and three angular velocities, viz. 36 r.p.m., 10 r.p.m., and 1^ r.p.m. The small cylinder was four diameters, and the other two cylinders were two diameters long. The observed velocities show a departure from theory, which increases with angular velocity and with the size of the cylinder, but which tends to vanish at very low speeds. The discrepancy is much greater * Communicated by the Author. Phil Mag. S. 6. Vol. 44. No. 259. July 1922. B 2 Mr. A. R. McLeod on Unsteady Motion produced for the " stopping " than for the " starting " experiments. The work was done at the Royal Aircraft Establishment, Farnborough, during the months Feb.-Sept. 1919. In some later work, not yet published, the discrepancy in the case of the " starting " experiments is traced to the effect of the base ; and the theory will probably apply to this motion in very long cylinders, provided eddies do not form owing to initial instability of the water. In the case of the " stopping " experiments, the discrepancy is due to instability and turbulence. § 1. Theoretical. In dealing with a rectilinear two-dimensional eddy in an incompressible fluid which contains no sources or sinks, the usual assumption is that particles of the fluid move in circles about the axis of the eddy. This makes the problem one of complete symmetry, and the radius vector r and the time t are the two independent co-ordinates. The equations of motion, when written in cylindrical co-ordinates with the axis of z along the axis of symmetry, reduce to tlie following forms, in which p denotes pressure, p is density, v is the kinetic viscosity, and <fi is the angular velocity about the axis : — ^ = p^' 2 (i) for the pressure, and dr* r "dr v ~dt (2) giving the angular velocity. Let us suppose that the angular velocity <j> satisfies the conditions = F(r) for t= 0, . . (3) </> = <f)(t) „ r — c = radius of cylinder ; . (4) that is, at the initial instant the angular velocity in the cylinder is known to be F(V) at radius r, while thereafter the rotation of the boundary is prescribed to be <j>(t). The solution of (2) satisfying the conditions (3) and (4) is, in by Change in Angular Velocity. 3 terms of a series of Bessel functions : «=i ' i'' o v a «y / Jo -j W,(«W«)«-"^ fWy,.^, . (5) a=] rJ (a„) f ${r)e va n* T l*dT, where J 1 (a )l )=0 and u n is the nth root of this Bessel function of the first order. Taking </>(£) =Ii = constant and F(r) = 0, we have O, r n=1 ««J 2 («yi) (6) This is the solution for the case in which the water is initially at rest and the cylinder suddenly rotates with a constant angular velocity fl. The solution is given as a problem in Gray & Mathew's ' Treatise on the Bessel Functions ' (Ex. 38, p. 236, 1st edn.). Taking </>(£) = and F(r) = XI = constant, we find the solution (j> _2c S Ji(* n r/c) H , C 2 12 ?< w= i u n 2 {tx n ) This is the solution for the case in which the water is initially rotating with a constant angular velocity O, and the cylinder is suddenly stopped. The solution is given by Stearne, Q.J. Math. xvii. p.' 90 (1881), and Tumlirz, Sitz. d. k. Alcad. in Wien, lxxxv. (ii.) p. 105 (1882). The phenomena which (6) and (7) are supposed to represent are at the basis of the formation and dissipation of eddies by viscous action. To take one example, they may be of importance in the theory of the aeroplane compass. The experiments of Part I. were undertaken to test the validity of these equations. § 2. -Numerical Solution of Equation (7), and Discussion. The numerical solution of (7) is given in Table I. for the values of r/c and vt/c 2 which are there indicated. The values are probably accurate throughout to the fifth significant figure. For small values of the arguments, the value of <j>/Q sometimes differed from unity only after the sixth figure. The values of J 1 (« /i r/c) for arguments greater B2 Mr. A. R. McLeod on Unsteady Motion produced Table I. Values o£ <j>/£l. l >t/c\ r/c = 0. •1. •2. •3. •4. •001 1-00000 1-00000 1-00000 1-00000 1-00000 002 1-00000 1-00000 1-00000 l-ooooo 1-00000 003 1-00000 1-00000 1-00000 1-00000 100000 004 1-00000 1-00000 1-00000 rooooo 1-00000 006 1-00000 1-00000 1-00000 1-00000 1-00000 010 1-00000 1-00000 1-00000 •99999 •99992 015 rooouo 1-00000 •99996 •99969 •99795 020 •99991 •99982 •99935 •99730 •98970 025 •99907 •99860 •99650 •99002 •97242 •030 •99587 •99451 •98923 •97591 •94634 •040 •97490 •97062 •95616 •92675 •87504 •050 •92923 •92196 •89890 •85665 •79077 •060 •86299 •85373 •82526 •77594 •70409 •070 ■78497 •77482 •74419 •69279 •62086 080 •70314 •69294 ■66248 •61234 •54393 •090 •62300 •61326 •58440 •53746 •47444 •100 •54776 •53877 •51224 •46943 •41257 ■150 •27221 •26753 •25321 •23045 •20079 •200 •13156 •12916 •12216 •11104 •09661 •300 •03034 •02979 •02817 •02560 •02226 400 *-0 3 6989 •Q 2 6862 •0 2 6489 •0 2 5897 •0 2 5128 •500 •0 2 1610 •0 2 1581 •0' 2 1495 •0 2 1358 •0 2 1181 •600 •0 3 3708 ■0*3641 ■0*3443 •0*3128 3 2721 •700 •0 4 8542 •0 4 8386 •0 4 7930 •0 4 7206 •0 4 6267 vt/c 2 . r/c = -5. •6. •7. •8. •9. •001 100000 1-00000 rooooo •99999 •97034 •002 1-00000 1-00000 1-00000 •99782 86681 •003 1-00000 1-00000 •99982 •98630 •76998 •004 1-00000 •99998 •99865 •96468 69193 . -006 •99999 •99944 •98952 •90552 57793 •010 •99886 •99004 •94263 •78057 44051 ■015 •98921 •95570 •85956 •65615 34305 •020 •96580 •90409 •77540 •56140 28162 •025 •93062 •84548 •69887 •48810 23855 •030 •88785 •78592 ■63137 •42973 20630 •040 •79277 •67418 •52033 •34229 16189 •050 •69768 •57707 •43399 27956 12940 •060 •60940 •49443 •36541 •23215 10650 •070 •53008 •42430 •30983 •19499 08892 •080 •45994 •36468 •26406 •16509 07497 •090 •39845 •31382 •22589 •14057 06265 •100 •34483 •27031 •19375 •12017 0.H30 •150 •16624 •12909 •09175 •05653 02544 •200 •07985 •08190 •04393 •02704 012155 •300 •018395 •014256 •010114 •0 2 6223 022797 •400 •0 2 4237 •0 2 3284 •0 2 2330 ■0*1433 3 6443 •500 •0 3 9760 •0 3 7563 •0 3 5366 •0 3 3301 3 1484 •600 •0 3 2248 •0 3 1742 •0 3 1236 •0 4 7605 4 3418 •700 •0 4 5178 •0 4 4013 •0 4 2847 •0 4 1752 5 7874 * -0-0989 means -C06989. by Change in Angular Velocity. 5 than 15 were obtained by the use of the first three terms of the asymptotic expansion. For smaller arguments the values were found by interpolation from the twelve-figure tables in Gray & Mathew's ' Treatise.' Curves representing Table I. are shown in fig. 1, from which figs. 2 and 3 have been derived graphically. The calculation was made 1-0 ■9 ■8 7 \m Fig.S. \\ ft 6 w v \ M) r^ — 1 \ ^ \ \ \\ \ \ \ IK ^ K XvC V 4 V X \\vV L O^M 8 \ X \\\^ %-^s p s ^^7 v ^^^r >, ^v^ "^h^*«-*L, "i" — f^^irr*'""- O .02. 04 .06 .12 J 4- .Ut. .16 J8 .90 with the aid of Chambers's seven-figure logarithms. It was checked throughout, once everywhere and twice or three times in parts. For large values of t, the angular velocities are pro- portional to Ji(ai?')/r {0 <r<_l). Most of the experimental curves are roughly of this form for a large value of t. The solution, being non-dimensional, applies to all sizes 6 Mr. A. E. McLeod on Unsteady Motion produced of cylinder, all angular velocities X2, and all incompressible fluids. The significance of vt/c 2 is of interest. According to (6) and (7), the behaviour of a cylinder of liquid is exactly similar to that of a cylinder of double the radius in four times the time. Larger eddies should therefore be of relatively longer duration than smaller eddies. If in these same cylinders we have different liquids, the liquid in the larger cylinder must have a viscosity four iimes that of the liquid in the smaller cylinder, in order that (f> may have the same value for the same values of X2, r/c, and t. For air i> = 0*14, and for water ^ = 0'011 at ordinary tempe- rature, the ratio being about 12. Ignoring the compres- sibility of air, eddies of the assumed kind should die away much faster in air than in water, the values of c and 12 being the same. According to (6) and (7), the rates of growth and decay of (£/X2 in the cylinders are independent of 12. We shall see from the experiments that this is true only when 12 is very small, or when the radius c is a small fraction of the length of the cylinder. § 3. Experiments. In the experiments which have been made, a brass cylinder, bored as accurately as possible and at least two diameters in length, was rotated about a vertical axis at constant speed. The cylinder was filled to within 1 cm. of: the top with ordinary tap-water ; and the observations consisted in timing through a measured angle by stop- watch, lycopodium particles floating on the surface, and so deducing the angular velocity <jf> at the radius r/c selected. The time of the stop-watch observation was also noted on a watch, in order to get the value of t from the instant of starting or stopping the cylinder. To time the lycopodium particles with more accuracy, a horizontal plane glass plate was mounted just over the cylinder. This was marked with ink in circles centring on the axis of rotation. The radii of the circles had the values r/c =0*3, 0*5, 0*7, and 0*9, and a fifth circle was added at r/c= l'O to aid in centring the plate over the cylinder. Straight lines, 45° apart, through the centre of the circles served to indicate the angles through which the particles were timed. A large plane mirror, inclined at 45°, was placed on the glass plate to enable observations to be made from the side. To obtain freedom from vibration, the driving-motor and by Change in Angular Velocity. 7 reduction gears were mounted on a separate table. The cylinders were clamped against a spinning-table which rotated very freely and easily in ball-bearings. Slight want of truth in the centring of the upper end of the cylinder was corrected by hanging a small weight on the rim, or by means of adjusting screws which worked in the rim of: the water-bath and pressed against the cylinder. Water-baths, providing a 2-inch layer of water around the sides and base, were fitted to each of the cylinders to make the temperature changes less. These rotated with the cylinder. In the case of: the larger cylinders especially, it was found that without a bath it was im- possible to secure uniform rotation of the water. When the room temperature was rising in the mornings or early afternoons, inward convection currents at the surface carried an excess of angular momentum towards the axis, and the angular velocity at the centre was sometimes for several hours 30 per cent, greater than that of the cylinder itself, the latter being very small. When cooling down, the effect is reversed and the "core" rotates slower than the cylinder. The water-baths made a great improvement, but with the largest cylinder, especially, the water in the bath and in the cylinder had to be mixed thoroughly before each experiment ; and the temperature of the water was regulated with an electric heater to be about o, l C. in advance of the room temperature (when rising). Some experiments lasted for nearly an hour, and a difference of o, l 0. between the temperatures of room and water was enough to start thermal currents having considerable effects (cumulative) on the observations. The cleaned surface of the water was lightly dusted with lycopo Hum, " rafts " not being allowed to form. When the temperature was being adjusted with the electric heater, the water was thoroughly mixed by stirring and bubbling air through it. The mean temperature was obtained for each experiment to 0°-01 C. At the end of each observation the stop-watch reading (given to 0*1 sec.) was recorded with the angle through which the lycopodium particle had rotated, and the time of the end of the observation as indicated by watch was also written down. A.11 observations in one experiment were made on a selected circle. The cylinder was timed frequently to verify the speed, which was constant to within a per cent, or two, and irregularities were corrected by an adjustable resistance. 8 Mr. A. R. McLeod on Unsteady Motion produced The motor was driven from a 200-volt accumulator circuit. Three sizes of cylinders were used : Small cylinder c = c x = 2*40 cm. = radius. Middle „ c = e 2 = 7'48 „ Large „ c = c 3 = 12*69 „ Three angular velocities were selected for each cylinder : Low speed £2 = 12! = '1396= 1^ r.p.m. Middle „ n=n 2 =ro47 =10 „ High „ O = H 3 = 3-770 =36 „ The large and middle cylinders were painted inside. They were bored with an error of a small fraction of a millimetre at the upper rim. The small cylinder was true to 0'05 mm., and its length was four diameters. The others were slightly more than two diameters long. The bases of all were plane and smooth on the inside. Experiments were made when the water began to rotate from a condition of rest, and also when it was coming to rest from a condition of uniform rotation, the uniformity being ascertained before the cylinder was stopped. In reducing the observations, the watch- reading was corrected by subtracting half the stop-watch reading to give the mean time t, and the values of vtjc 2 and 0/X2 were calculated for each observation. The viscosity of the water, at the mean temperature during the experiment, was taken from Kaye & Laby's Tables. The values of 0/12 were plotted on squared paper against the values of vt/c 2 , each observation being represented by a dot. Hence for a selected circle corresponding to the selected value of r/c, for each value of 12 there corresponds a series of dots which lie in a narrow band. This band defines a curve by its median line. Observations were made on the four circles corresponding to r/c = 03, 0*5, 0*7, and 9 for each of the three cylinders. Thus for each cylinder there were finally eight sheets of curves — four for " starting " and four for " stopping- " experiments. Each of these sheets contained three curves corresponding to the three selected velocities, and corresponded to one of the values of r/c. The theoretical curve from Table I. was also added for comparison. Several experiments were made for each set of conditions, to secure a sufficiently dense band of points for each curve. Apart from a number of preliminary by Change in Angular Velocity. 9 experiments, the results given here are based on 215 distinct experiments. In taking the stop-watch readings, disturbances of the circular motion of the lycopodium owing to turbulence or eddying were avoided as much as possible, and selections were made of those motions which lay most nearly along the circle considered. This means that when turbulence is increased, the angles through which the particles are timed become less. The effect of the turbulence is merely to broaden the band of points, but the mean motion, repre- sented by the median line, is always well-defined. § 4. Discussion of Results. The results obtained are embodied in the accompanying curves. These are derived from the observational curves mentioned in § 3. The following reference table explains the figures : — Stopping. Starting. Stopping. Starting. Stopping. Starting. Stopping. Starting. Stopping. Starting. Stopping. Starting. Stopping. Starting. Stopping. In these figures each curve represents the distribution of angular velocities </}/£! over a radius of the cylinder at the time vtjc 2 corresponding to the number given alongside the curve. The times in seconds corresponding to these numbers may be obtained by use of the following table, the water being at 17° C. : — c l = 2'40 cm. vtjc 2 = '01 corresponds to /= 5*30 sees. c 2 = 7-48 „ vt/c 2 = -01 „ „ t= 51-8 „ c 3 =12-69 „ vt/c 2 = '()l „ „ t= 148-4 „ .4. Cl =2-40 cm. n 2 =10 r.p.m. 5. •>•> I2 3 = 36 6. ?5 £2 3 = 36 „ 7. c 2 = 7*48 cm. n t = n „ 8. >y n,= ij „ 9. 55 G 2 =10 „ 10. ;? n 2 =io „ 11. 5> n 3 =36 12. ?5 X2 8 =36 13. c 3 = 12-69 cm. Gi= H » 14. ?i n,= ii „ 15. ?5 n 2 =io 16. •>y n 2 =io 17. 5? n 3 =36 18. J? 123 = 36" 10 Mr. A. R. McLeod on Unsteady Motion produced Comparison with the theoretical curves (figs. 2 & 3) shows a marked departure in all cases but that o£ the slow speed and small cylinder. The agreement with theory improves as r—s~c, but there is still a large departure at r/c = '9 for the large and middle cylinders, except at the low speed. Vc w Again, it is noticeable that as the radius of the cylinder increases, the departure from theory becomes more marked. This might be expected as there is, near the axis, relatively less constraint from the boundaries with the larger cylinders. Accordingly, if we suppose an eddy of this kind rotating in a lake of stationary water, and if instead of stopping the cylinder wall we annihilate it, we expect the eddy to dis- appear more quickly than if the stationary solid wall had by Change in Angular Velocity. 11 been retained, for the constraint will be still further reduced and greater irregularity is possible. It is particularly noticeable that as tlie constant rotation D. of the cylinder is increased, the departure from theory becomes more marked. Only three figures were obtained for the small cylinder, because the agreement with the theoretical curves in the three cases omitted will be practically exact. The curves for the " stopping " experiments show a much greater departure from theory than the curves for the "starting" experiments. This is due to the break-up of the regular motion, owing to instability at the fixed outer wall. Except with the high speed and the large cylinder, 12 Mr. A. R. McLeod on Unsteady Motion produced the motion on starting, on the other hand, appears to the eye to be without appreciable irregularity, and it is very striking to see the sharp dividing-line between quiescent liquid in the centre and rotating liquid on the outside. This dividing-line, represented by the steep part o£ the curve in the figures, slowly moves towards the centre but becomes indistinct some distance from it. It is best seen at the higher speeds when the velocity-gradient is greater. Its rate of travel depends on the value of O. In the case of the large cylinder starting at the high speed, large secondary eddies 3*5 cm. across were often observed just inside the cylinder wall a few seconds after starting the motor. These soon died out when the velocity-gradient became less, and thereafter the motion travelled in towards by Change in Angular Velocity. 13 the centre regularly. The effect of these eddies was observed to cause a shifting of the curve for <j>/£l in the direction of a greater <£ for the same value of vtjc 2 . In the same case, secondary eddies were observed about the "4 circle when the motion had reached the centre. Although mean velocities were recorded (median Hue of the band of points), the effect is shown in fig. 17 by the wavy appearance of tie two upper curves. When the cylinder is stopped, the water continues to rotate until the irregular motions, generated near the cylinder wall, have had time to extend inwards. Small eddies then travel about, and the central axis of rotation wanders con- siderably and often seems to disappear temporarily amid cross-currents. The motion is very irregular except at the low speed, and even in this case some irregularity al\\ ays remains. The lycopodium particles do not follow the circles for very long, and are usually moving at an angle to them. With the large cylinder at the high speed, the velocity immediately after stopping the cylinder seemed to give stability and to aid in preserving the circular character of the motion ; but when the kinetic energy had somewhat diminished, eddying became more noticeable. On the curves mentioned in § 3 in which <£/H is plotted against vtjc 2 , the bands of points are much narrower in the " starting " experiments than in the others, and determine the position of the median line easily to *001 in the value of vtjc 2 in most cases. For the " stopping" curves, the limit of error may be two or three times this occasionally. One noticeable effect is that the band is narrow when the velocity- gradient has a considerable value, i. e. when the curves in the figures slope steeply. In these cases considerable momentum is being transferred through the water, and there will be considerable shearing stress and vorticity, and the stability might therefore also be considerable. As soon as the velocity-gradient becomes small, the band of points broadens. For example, in the " starting " curves the bands are some- times very narrow until the value of <j>/fl has risen to 0*9, when they broaden out. Conditions seem to favour irre- gularity at the centre (axis) of the cylinder where the velocity-gradient vanishes. On the axis the stability is a minimum. In the "stopping" curves the bands are nar- rower the greater the angular velocity O, i. e. the greater the vorticity of the water, especially near the cylinder wall where the instability originates. The observational curves show that viscosity alone is not 14 Unsteady Motion in a Rotating Cylinder of Water. sufficient to account for the effects, except for small values of c and Q, i. e. for long, narrow cylinders and slow speeds. A little dye introduced into the rotating water shows no signs of any minute eddying or micro-turbulence ; and so we must look for currents in the water as the cause, of the discrepancy, which is obviously the case in the " stopping " experiments. The formation of large eddies in large bodies of fluid seems to be due chiefly to the interaction of two local currents, or to low pressure caused by an obstacle or a sink, and not to the slower processes of viscosity. If we attribute the deviation from theory to an ignored increase in the kinetic viscosity v, we find that when the large cylinder is stopped at the high speed, the increase would have to be represented by a factor exceeding 10 in value nearly everywhere, while the value would lie between 50 and 100 on r/c = 0'3 shortly after stopping. With the middle cylinder, stopped at the high speed, the factor has about half these values ; and with the small cylinder, stopped at the same speed, the factor ranges from 1*3 to 3*0. In the starting experiments the factors are nearly unity, but they are meaningless here as the motion is not turbulent. Some earlier experiments illustrate the instability of the stopping experiments. In these an inner cylinder was rotated coaxially with a fixed outer one. As is well known, it was found that at no speed of rotation of the inner cylinder was it possible to set the water moving in circular paths, o wing- to the eddies which were continually thrown off. The slower the speed of rotation the more conspicuous were the eddies, especially on the borders of the outer, more slowly-moving water. Measurements of the angular velocity showed a large departure from theory, the inner parts rotating more slowly and the outer parts more rapidly than the theory indicates. The effect o£ the travelling eddies is thus to make the angular velocity more like that of a rigid body. When the speed was very great (2500 r.p.m.) the kinetic energy seemed to give stability to the water. A whirlpool formed next the inner cylinder, and a large oscillation was presently set up in the form of a wave with its crest along a radius of the outer cylinder and its trough on the other half of the same diameter. Some thick, very viscous oil residues, when rotated in a cylindrical tin about 15 cm. in diameter, acquired the full velocity on starting (36 r.p.m.) in something less than 4 seconds, and came to rest in the same time when the cylinder was stopped. Only a slight displacement of the oil occurred, the surface being momentarily roughened with fine lines like cracks. [ 15 ] II. The Elements of Geometry. By Norman Campbell, Se.D* Summary. IT is maintained that the geometry of Euclid is best interpreted as an attempt to deduce as many important propositions as possible from the assumption that length, angle, area (and perhaps volume) are magnitudes uni- versally measurable by the methods that are actually employed in experimental physics. All his chief pro- positions (in so far as they are true) can be deduced from that assumption without any other. This view is supported, not by a detailed analysis of the Elements, but by a very summary sketch of the laws that must be true if the assumption is to be acceptable. In a sequel it is hoped to discuss similarly the foundations of another branch of experimental geometry with which Euclid is not directly concerned — namely the geometry of position, which involves the concept of " space." 1. There was formerly much discussion whether geometry was an experimental or a mathematical science. It is now generally agreed that there are two closely connected sciences, one mathematical and one experimental. The former, which has been defined as the study of multi- dimensional series, consists of a logical development of ideas which have no necessary dependence on the experience of the senses. It does not consist of laws and cannot be proved or disproved by experiment ; it can enter into relation with experimental science only through theories and by suggesting hypotheses which, interpreted suitably, predict laws. The formulation of such theories, in which Minkowski was the pioneer, is one of the most striking features of modern mathematical physics. The experi- mental science, on the other hand, is meaningless apart from experience, and its propositions are true or false according as they agree or disagree with experiment. They are the very fundamental laws which involve only the geometrical magnitudes such as length, angle, or area. It may be noted in passing that the laws predicted by geometrical theories are not in general geometrical laws, but involve electrical, optical, or dynamical concepts. * Communicated by the Author. 16 Dr. Norman Campbell on the The mathematicians who have recently taken over from the philosophers the task of teaching experimenters their business have decided that only the mathematical science is properly termed geometry. In support of their claim they appeal to the authority o£ the Greeks, and thereby imply that Greek geometry is mathematical and not experi- mental. This implication raises questions of scientific interpretation and not of mere convenience in nomenclature. For the matter cannot be decided by inquiring what Euclid (for example) thought he was writing about : it is admitted that, as an exponent of mathematical geometry, he was guilty of errors ; and, if he was capable of error, he may have been wrong as to the nature of his assumptions and of his arguments. If we are justified today in confining the term to one study rather than another, because that term was used by Euclid, it can only be on the ground that Euclid's propositions and his methods of proving them are closely similar to those employed today in that study. If this test is applied, geometry is an experimental science. For whereas the Elements is utterly different from anything modern mathematical geometers produce, it is, judged by modern standards, quite a creditable attempt at an exposition of experimental geometry. It can be regarded broadly as an attempt to deduce as many important laws as possible from the single assumption that length, area, angle, and (less definitely) volume are magnitudes, universally measurable by the methods which are actually employed in experimental physics, or to which the methods that are actually employed would be referred if doubt arose concerning their validity. Nothing is assumed but that every straight line has a length, every pair of straight lines an angle, and every plane surface an area. The definitions, axioms, and postulates should then be state- ments of the laws by virtue of which measurement is possible. It is admitted that the attempt is not wholly successful ; but its faults, or many of them, are readily explicable : the author has not to be represented (as he must be if he is an exponent of the mathematical science) as constantly straining at gnats and swallowing camels. Such a view can be established only by a detailed and tedious criticism which, in so far as it concerns Euclid's intelligence, is not of scientific interest. In place of it will be offered a very summary sketch of the fundamental notions and laws of experimental geometry and sufficient comparison of them with Euclid's assumptions to suggest that on them might be founded a deduction, by methods Elements of Geometry. 17 very similar to those that he employs, of the propositions which he actually states. References are throughout to Todhunter's edition. (2) But two preliminary questions must be asked. F-irst, can an experimental science be deductive at all? Certainly it can. A deduction from a law is an application of that law in particular circumstances which were not examined when it was formulated. If, after examining the sides of squares and of triangles, I assert the general law that all straight lines have measurable lengths, and then, without further experiment, assert that the diagonals of squares, which are also straight lines, are also measurable, I am miking a deduction. It may be true that there is some- thing precarious about the results of such deduction — that question is not raised here, — but the deduction itself is quite unexceptionable ; the falsity of the conclusion is definitely inconsistent with the truth of the premises. If deubt is raised concerning the conclusion, the ultimate means of resolving it is by experiment ; but experimental science, in the hands of its greatest exponents, consists in asserting- such general laws that doubt does not arise concerning the results of deduction based on them. The second question is whether there are truly laws which make measurement possible. The question is dis- cussed at length in my ' Physics/ Part II., the results and nomenclature of which will be used freely in what follows. But there is one matter which may receive special mention here, because it is concerned with " incommensurables," which are often (but falsely) believed to be of especial importance in geometry. Measurement is possible when, by means of definitions of equality and addition, a standard series of the property in question can be established, starting from some arbitrary unit, such that any system having the property is equal in respect of it to some one member of the standard series. Now (it might be argued) such mea- surement is not possible for length, because the diagonal of a square cannot be equal to any member of a standard series based on the side as unit ; indeed that result is actually proved by Euclid. Consequently it is patently absurd to pretend that Euclid's propositions can be derived from an assumption, namely that measurement is possible, which is inconsistent with its conclusions. One method of escape from this difficulty may be mentioned, although it will not be adopted. A slight Phil. Mag. S. 6. Vol. U. No. 259. July 1922. J 8 Dr. Norman Campbell on the amendment in the thesis might be made, and it might be said that Euclid's assumption is that the laws are true which would make measurement possible if there were no incommensurable lengths — for these laws, though necessary to measurement, may not be sufficient. But the difficulty vanishes entirely, if it is remembered what is meant by " equality " in experimental measurement. When it is said that A is equal to B, it is meant that there is no possible means of deciding which of the two is the greater. If then I say that the diagonal of a square is \/2 times the side, I mean that, if I measure the diagonal in terms of the side as unit, there is no means of deciding whether the value obtained, when multiplied by itself according to the multiplication table, will be greater or less than 2. That statement is not in the least inconsistent with my assigning to particular diagonals values of which the square is not 2 ; it is only inconsistent if a law can be found by which I can tell in particular cases whether the square will be greater or less than 2. My assertion is that* there is no such law ; and that assertion is true. In its appli- cation to all magnitudes except number, equality must be interpreted in this, slightly statistical, sense. 3. There is then no preliminary objection to the view that Euclid's propositions are deductions from the laws in virtue of which the geometrical magnitudes are measurable. We now proceed to ask what those laws are. Geometrical conceptions are derived ultimately from our immediate sensations of muscular movement, just as dynamical conceptions are derived from our sensations of muscular exertion and thermal conception from our sense of hot and cold. We have an instinctive and indescribable appreciation of differences in direction of various movements ; we appreciate that one direction may be between two others ; and if other sensations (e. g. those of hot and cold or rough and smooth) vary with movement along a certain direction, we appreciate that of the varying sensations some are between others. The notions of direction and of the two kinds of betweenness are the foundations of geometry. It is a vitally important fact that there is an intimate relation connecting betweenness determined by one kind of muscular motion (e. g. that of the hand) and that determined by another {e.g. that of the eye). The relation is much too complex for any account of it to be attempted here ; but it is only because it exists that "space" explored visually or by our different limbs is always the same. Elements of Geometry. 19 The fundamental notions give rise to those o£ surfaces and lines. Surfaces are connected with the fact that a sensation may be unaltered by movement in any of a certain group of directions (which are said to be in a surface and to cha- racterize it), while it may be altered by any movement in any direction not in this group (directions away from the surface) . Of lines there are two kinds, which will be termed respectively " edges" and " scratches."" Edges arise from the fact that the group of directions characteristic of a surface may change suddenly at some part of it. It is a matter of convenience whether the parts characterized by different directions are spoken of as different surfaces or as parts of the same surface : we shall adopt the second alternative. Scratches arise from the fact that, while the directions characterizing a surface are unaltered, the sen- sation the occurrence of which distinguishes "in the surface" from "out of the surface" may change suddenly. Some, but not all lines, are such that the whole of them lies along a single direction. Points are of little importance in the earlier stages of geometry ; they arise from the fact that two lines may have a part in common. Two points, both on the same line, are termed the ends of the part of the line between those points. The recognition of surfaces and lines is the first step towards geometry. Euclid attempts to give an account of them in Defs. 1, 2, 3, 5 of Book I., which are the least successful part of his treatise. The account given of them here is no better than Euclid's for the purpose of conveying a notion of them to one who does not possess it already ; but since there are no such persons, the objection is not serious. But our account is better in drawing attention to the notions that are fundamental in geometry and in not assuming familiarity with conceptions, such as length, which are necessarily subsequent. 4. Some surfaces, but not all, when subjected to muscular force undergo only such changes as can be compensated by a suitable movement of the whole body ; if such a movement is made, the group of directions characterizing the surfaces is restored. In other words, such surfaces can move without alteration of form ; they provide the original and crude con- ception of a rigid body. By means of the motions of rigid bodies, it is sometimes possible to bring parts of two pre- viously distinct surfaces info contiguity, so that (here is nothing between those parts. In particular, edges, or parts of edges, can often be brought into such contiguity. Scratches 02 20 Dr. Norman Campbell on the can be brought into contiguity with edges, and, in a sense, into contiguity with other scratches ; but the criterion of contiguity in the last case is much less direct and requires methods involving something other than the simple per- ception of nothing between. The recognition of the possibility of contiguity is the second step towards geometry and leads immediately to the third, which consists in the establishment of a definite criterion for a straight line. A crude criterion is provided by direct perception : a young child knows the difference between a straight and a bent line by simply looking at them; the recognition seems to depend on the fact that a straight line is all in one direction and is symmetrical with regard to the unsymmetrical directions of left and right or back and front. The crude criterion is stated as well as it can be in Euclid's Def. 4. But contiguity provides a much more stringent criterion, which in the first instance is applicable only to edges and not to scratches. Two edges are straight if, when two portions of one are brought into contiguity with two portions of the other, all the portions between these two portions are also in contiguity, however the contiguity of the first pairs of portions is effected. It appears as an experimental fact, that if A, B and 0, D are two pairs of straight edges according to this criterion, C is also straight if tested against A ; accordingly an edge can be called straight independently of the other member of the pair on which the test is carried out. A scratch is straight if it can be brought into complete contiguity with a straight edge, Those facts are stated in Axiom 10. Other definitions of a straight line are sometimes offered : e. g., (1) an axis of rotation, (2) the shortest distance between two points, (3) the path of a ray of light. (1) is almost equivalent to that stated here; (2) will be noticed presently ; (3) is not accurately true (i. e., if it is adopted, the familiar propositions about straight lines are not true), but it is important as an approximation for comparatively rough measurements. A plane surface (or, according to our usage, part of a surface) is then defined as in Def. 7. It can also be defined by the complete contiguity of three pairs of sur- faces ; but the contiguity of surfaces is not easy to describe accurately. Such a definition is, however, actually used in making optical flats and surface plates ; if it were adopted, it would still be necessary to introduce the fact that it agrees with our definition, in order to measure angle. The 7<?r Elements of Geometry. 21 conception of the contiguity of surfaces is not actually required, except perhaps for the measurement of volume. {Cf. § 11.) 5. The third step places us in a position to introduce measurement and the three fundamental magnitudes, length, angle, and area. For fundamental measurement we need definitions of equality and addition, such that the law of equality and the two laws of addition are true. The choice of unit may be left out of account ; for, with geometric magnitudes, the laws are true whatever unit is selected. The law of equality is Axiom 1; the first law of addition is Axiom 9. Axioms 2-7 are together very nearly equivalent to the second law of addition (which may be stated roughly in the form that the magnitude of a sum depends only on the magnitudes of the parts). Axiom 8 is an attempt to compress the definitions of equality for all three magnitudes into a sinole sentence ; it is better to separate them. Euclid fails to give any definition of addition ; he does not tell us how the " w 7 hole " is to be related to the "parts" in order that it should be greater. 6. We will now take the magnitudes in turn. For the length of a straight line the necessary definitions are : — (1) Two straight lines are equal in length if they can be placed so that when one end of the first is contiguous with one end of the second, the other ends are also contiguous. (2) The length of the straight line AB is equal to the sum of the lengths of the straight lines CD, EF, if they can be placed so that C is contiguous with A, F with B, I) with E and with some part of AB between A and B. These definitions, like all similar definitions of mag- nitudes, are satisfactory and are subject to the necessary laws of equality and addition only if certain conditions are ful rilled. The conditions are described by saying that the surfaces in w T hich the straight lines lie must be those of rigid bodies. This is a definition of a rigid body: a rigid body is something which (like a perfect balance) is determined by the satisfaction of the conditions for mea- surement *. Rigid bodies according to this test include many of those which satisfy the crude test of § 4, though they include others (e.g., surveyors' tapes used as surveyors use them) which do not satisfy that test. In virtue of the fact that rigid bodies are necessary to measurement, the * Cf. H. Dingier, Phys. Zeit. xxi. p. 487 (1920). 22 Dr. Norman Campbell on tlie branch of geometry with which we (and, according to our view, Euclid) are concerned may be fitly described as the study of the surfaces of rigid bodies. It is thus dis- tinguished from a wholly different branch of geometry, with which we are not here concerned, ihat is not confined to rigid bodies ; this is the geometry of position. It is important to notice that not all pairs of straight lines can be brought into contiguity, and that the law of equality cannot therefore be tested universally. It might have turned out that there was some material difference between those which can and those which cannot be brought into contiguity with a given line ; and that if we assumed that the law of equality is universally true, we should be led to inconsistencies. It is an experimental fact that no such inconsistencies do arise when we extend our definition of equality so that lengths are equal when they are equal to to the same length, although they cannot be brought into contiguity with each other. This is, of course, one of the most important laws that make measurement possible. A similar remark applies to all the geometric magnitudes and need not be repeated. 7. The length of lines that are not straight can be measured approximately as fundamental magnitudes by means of flexible but inextensible strings. But the laws of such measurement are not strictly true, because (as we say now) no string is infinitely thin and the surface never coincides with the neutral axis. Another possible way, perhaps more accurate but of limited application, would be to roll curved edges on some standard edge, which need not be straight. But in truth there is no perfectly satisfactory way of measuring fundamentally the length of curved lines. All the measurements which we make on them are derived from measurement of straight lines ; they involve numerical laws between fundamentally measured magnitudes. One of these laws is that the perimeters of the circumscribed and inscribed regular polygons tend to a common limit as the number of sides is increased. That law is therefore a law of measurement if curved lines are to be measured. The question whether curved lines can be measured fundamentally is important, because, if they could be, it would be possible to define a straight line as the shortest distance between two points. (The definition would have to be put in some other form, since distance, a conception belonging to the geometry of position, implies the mea- surement of length.) But since they cannot be, that Elements of Geometry. 23 definition must be rejected ; it must be regarded merely as a generalized form of Prop. I. 20. 8. Angle is the measure of the crude conception of direction. The following are the definitions of equality and addition for the angle between two intersecting straight lines : — The angle between two straight lines A, B is equal to that between C, D if it is possible to bring A into con- tinguity with C and B with D. The angle between A, B is the sum of the angles between C, D and E, F, if, when A is brought into contiguity with C and D with E, I) lying between C and F and in the same plane with them, F can be brought into contiguity with B. These definitions are satisfactory only if the straight lines are in rigid bodies; or, in other words, there are surfaces which satisfy the conditions for the measurement of length and also those for the measurement of angle. But even if the surfaces are those of rigid bodies, the definitions are not wholly satisfactory and the laws of measurement not entirely true. We must distinguish angles according as the two straight lines which they relate are or are not prolonged on both sides of the common point : the latter class may be termed " corners/' the former " crossings." Angles between edges are always corners ; those between scratches may be either corners or crossings. If we try to include both corners and crossings in the same class as a single magnitude, the law of equality is not true ; for two corners which are both, according to the definition, equal to a crossing may not be equal to each other ; as we say now, one angle may be the supplement of the other. But if we treat corners and crossings as separate magnitudes this difficulty disappears ; the law of equality is true for either taken apart from the other. Actually we take corners only as magnitudes ; crossings we measure by the corners with which they can be made contiguous. Each crossing then has four angles (i. e. corners) associated with it. It is an important experi- mental fact that the " opposite" angles are equal; it is best taken as a primary law, instead of being proved from other axioms as in Prop. I. 15. It is a law of measurement, because if it were not known, we should need four and not two angles to measure a crossing ; it is thus inherent in our system of measurement. But though the law of equality is now true, the first law of addition is false ; it is false for both corners and crossings. The whole which is the sum of the parts may be equal to 24 Dr. Norman Campbell on the one of the parts : e. </., if both of two parts, being corners, are what we now call 120°. Some kind of spiral space can be imagined in which the law would be true ; but actually it is very important that it is false. For, apparently in- separable from its falsity is the fact that the angle between two portions of the same straight line can be measured and given a finite value in terms of a unit which is the angle between two intersecting lines. This fact is described by the assertion that there are right angles and that a per- pendicular can be drawn to any straight line from any point in it, a right angle being defined as in Def. 1. 10. (Axiom 1. 11 follows from this definition, regarded as an existence theorem, and our axiom Prop. I. 15.) Since the existence of right angles is vital to geometry, we cannot avoid the falsity of the first law of equality by some alteration of the definition. We can only recognize that the law is true in some conditions, and be careful to apply it in deduction only when it is true. It is true when all the lines making the added angles lie on the same side of (or contiguous with) a single straight line passing through their common point ; this condition can be expressed, though with some complexity, in terms of the fundamental notion of between. Thus, in proving Prop. I. 16 we need to know that OF and CD both lie on the same side of AC This law, and perhaps others of the same nature, are laws of measurement, defining the conditions in which angle can be measured uniquely. They require explicit mention. The ambiguity which the falsity of the first law of addition introduces into numerical measurement is removed by certain conventions. These need not be considered here for we are not assigning numerical values. If the length of curved lines were measurable funda- mentally, angle might be measured as a pure derived magnitude, e. g. by the ratio of the arc to the radius of a circle in virtue of the numerical law, established experi- mentally, that the arc is proportional to the radius. But since curved lines cannot be so measured, we must take angle to be fundamental. We cannot use right-angled triangles with straight sides to measure angle as derived, because we need fundamental measurement to determine what angles are right. Of course we might define for this purpose a right angle as an angle between some two lines arbitrarily chosen as standard ; but such measurement would be intolerably artificial and nothing whatever could be deduced from such a definition. Elements of Geometry. 25 9. Euclid's definition of parallel lines must be rejected entirely, for, since all plane surfaces are limited, the criterion suggested is inapplicable. Since the crude de- finition of parallelism is similarity of direction, we may try to define parallel lines as those which being in the same plane make the same angle with any third line. We thereby imply the axiom of parallels in the form (Prop. I. 29) that such lines which make the same angle with one straight line make the same angle with any other ; we imply also that the angles which are to be equal are the "exterior" and "interior" opposites or the "alternate" angles, since if the interior angles are compared the proposition is not true. But the definition is not very satisfactory; for, when the lines are edges, there is not always an exterior or an alternate angle. It is better to adopt the substance of Axiom I. 12 as a definition, and to say that lines in one plane are parallel when the sum of the interior angles is equal to two right angles. This much abused axiom seems to me a very ingenious way out of a real difficulty. We then assert the axiom of parallels in the form (implied by I. 32) that if any two straight lines in a plane are cut by any third line, the sum of the interior angles is the same for all third lines. The merit of this axiom is that it indicates clearly that the " axiom of parallels " is really something concerning all straight lines in a plane and not only parallel lines, and that parallel lines are merely a particular case of other pairs of lines. The propositions that parallel lines never do intersect and that the angle between them is zero follow immediately. The axiom of parallels is a law of measurement because it is involved in the measurement of the angle between lines w r hich do not intersect. Its use for this purpose requires that at some point of a straight line it should always be possible to place a straight line parallel to a given straight line. This proposition is not true for concave surfaces, but the complexities arising from this failure and the means of avoiding them may be left for the present ; they are dealt with more naturally in connexion with "space." If the axiom were not used, we could not by our present methods measure the angle between non-intersecting straight lines : first, because the definition of equality given above, though sufficient for such lines, is not necessary : second, because the definition of addition is wholly unsatisfactory. There has been so much discussion of the necessity of the 26 Dr. Norman Campbell on the axiom of parallels that the matter requires rather more con- sideration. Two questions are involved. First, would it be possible to measure the angle between non-intersecting lines without assuming some proposition logically deducible from the axiom? It would be if, and. only if, some property, common to all lines between which the angle is the same, can be found which is determinable by direct experiment not involving parallel lines. There may be such a property, but I have not been able to think of it. Second, if the axiom were not actually true — but we may stop there. In a pure experimental science, there is no sense in asking what would happen if the world were other than it actually is. Theory is necessary to give such a question a meaning, by suggesting what might remain unaltered during the change. For our present purpose the axiom is as necessary as any other of those we are considering. 10. Area is distinguished from all other fundamental magnitudes because the definitions of equality and addition are inseparable. They may be expressed thus. The areas of two bounded plane surfaces are equal if (but not only if) their boundaries can be brought into complete contiguity with each other or with the same third boundary. (A bounded surfaca is a part of a surface which includes all portions which can be traversed without crossing the boundary line.) The area of A is the sum of the areas of B and C, if when parts of the boundaries of B and C are brought into contiguity with each other, the remaining parts of the boundary can be brought into contiguity with the boundary of A. In virtue of the fact that parts of the boundaries of two surfaces can be brought into contiguity in many different ways, there may be many different bounded surfaces, of which the boundaries cannot be made contiguous, which are the sum of the same bounded sur- faces. If the measurement of area is to be satisfactory, these surfaces must also be deemed to have equal area, and the definition of equality must be extended correspondingly. With this extension the laws of equality and addition are true, and the measurement is satisfactory. In order that all bounded plane surfaces should have areas, some rule must be found for choosing the shape of the members of the standard series and for grouping them in such a way that some sum of them is equal to any area. We use for this purpose rules based on the axiom of parallels, and that axiom is therefore again a law of the Elements of Geometry. 27 measurement of area. The rule might possibly be dis- pensed with, it we were prepared to spend unlimited time in selecting by trial and error shapes for the members of the standard series which fulfil the necessary conditions ; but actually we could never measure area except by making use of similar figures, the production and properties of which depend wholly on the axiom of parallels. Further, it is the use of that axiom which enables us nowadays to calculate area from the linear dimensions of a surface without resorting at all to fundamental measurement. But of course all the numerical laws on which that cal- culation depends have to be established by means of fundamental measurement. It is only by defining area as we have done, and assuming the axiom of parallels, that we can prove by deduction that the area of a rect- angle is proportional to the product of its sides, or equal if the units are suitably chosen. The areas of surfaces that are not plane cannot be mea- sured fundamentally, even to the extent that the length of curved lines can be. For there are no inextensible surfaces which can be brought into contiguity with surfaces of any curvature. Measurement of curved area is always derived and estimated by the limit of the circumscribed polyhedra as the number of their sides is increased. But the whole matter is obscure, because it is much more difficult to establish experimentally that there is a limit or to say what the limit is ; for there is here no inscribed polyhedron tending to the same limit. There is singularly little experi- mental evidence for the assertion that the area of a sphere is 47rr 2 , and there is great difficulty in saying exactly what we mean by such an assertion ; curved area is almost always a hypothetical idea and not an experimental magnitude at all. 11. Volume is a property of complete surfaces. Since complete surfaces can never be brought into complete con- tiguity, volume cannot be measured fundamentally by any process at all similar to those applicable to the magnitudes we have considered so far. Volume is measured (1) as a fundamental magnitude by means of incompressible fluids, or (2) as a derived magnitude by means of the lengths and angles characteristic of the surface. The second method depends upon numerical laws established by means of the first. In certain cases these laws can be related closely to other geometric laws by means of the following propo- sitions : — (1) Two complete surfaces with equal dimensions, 28 Dr. Norman Campbell on the i. e. with equal lengths and equal angles between them, have equal volumes. (2) If two complete surfaces have each one part plane, and the boundary of the plane part of one can be brought into complete contiguity with the plane part of the other, then the complete surface which has dimensions equal to that of the complete surface so formed has a volume equal to the sum of the volumes of the original surfaces. These propositions could be used as definitions of equality and addition in a system of measurement, which would be independent of the measurement of length and angle (and therefore not derived), because it involves only equality, and not addition, of length and angle. But it is of limited scope and, in particular, would not permit the measurement of the volumes of curved surfaces. Since we do undoubtedly attribute a meaning to the volume of such surfaces, in a way that we do not to their area, measurement by incom- pressible fluids, which is not geometric, cannot be wholly avoided. But the propositions, which are those on which Euclid bases his treatment of volume, are actually used in modern practice, and are therefore regarded permissibly as laws of measurement. 12. In deducing Euclid's propositions from the laws of measurement of these magnitudes, subsidiary laws are required, corresponding roughly to his postulates, expressed and implied. First, we need "existence theorems" corre- sponding to each of the definitions; for example, the definition of a plane surface justifies the conclusion that a straight edge can be placed contiguously to any two portions of such a part of a surface. Second, we need the assumption that we can make an object having a magnitude equal to that of any object presented to our notice. All these propositions are laws of measurement : the first group, because ail definitions in experimental science are nothing but existence theorems ; the second, because it is implied in the fact that we can make a standard series by which we can measure any magnitude. Euclid's three expressed postulates are all untrue. I cannot " draw a straight line " from this room to the next when the door is closed. Moreover his constructional propositions, closely connected with the postulates, are unsatisfactory because they are all directed to the drawing of scratches, rather than to the making of edges. The hypothetical experiments by means of which the deductions are effected are carried out much more easily with edges Elements of Geometry. 29 than with scratches ; and if any of the propositions were doubted and put to the test of experiment, it would certainly be by means of edges ; the extension to scratches would be by means of the contiguity of edges with them. Euclid's methods here undoubtedly indicate that he is leaving, perhaps consciously, the realities of experimental science tor the pure ideas of mathematics. But he has made so little progress towards the new peak that, if he is to be restored to safety, it is far easier to drag him back to that which he lias never left completely than to guide him through the bog in which the two sciences are confused to the very distant goal. 13. Only a few disconnected remarks will be offered here on the process of deducing the Euclidean propositions from the fundamental laws that have been sketched. Of course, we should employ the " application " (or contiguity) method of Prop. I. 4 wherever possible, instead of trying to avoid it ; for it is based directly on the fundamental notions. Again, we should not commit Euclid's error of supposing that strictly similar triangles can be brought into contiguity; we should apply the mirror image first to one triangle and then to the other. There would be no need to introduce area to prove Prop. I. 47. A Greek writer was forced to do so, because, not being familiar with the multiplication table, he could describe in no other w r ay the relation between a number and its product by 'itself. We should proceed from Prop. I. 34 to Book VI. and prove Prop. I. 47 by drawing the perpendicular from the right angle to the hypoteneuse and using the relations of similar triangles, treated by algebra. For nowadays, since w r e admit no incommensurable magnitudes, we can dispense altogether with Euclid's very beautiful and ingenious subtle! ies about ratios. A ratio in experimental science is nothing but a value taken from the multiplication table, which is established by the measurement of number, i. e. by counting. The laws of the measurement of number are involved in those of the measurement of every "continuous" magnitude. April 22, 1922. [ 30 ] III. On the Hotation of Slightly Elastic Bodies. By Dorothy Wrinch, D.Sc, Fellow of Girton College, Cambridge, and Member of Research Staff, University College, London * . THE change in dimensions of a slightly elastic body due to rotation is a question of some practical importance, and does not appear to have received any systematic treat- ment. In the theory of elasticity, the displacements of a point of the body are of course discussed and the displace- ments of the points of the boundary determine the increase of dimensions. But the problems of elasticity which are of interest mainly from the point of view of increase of dimen- sions, rather than of the distribution of stress in the material, can rarely be solved by the current methods or appear onlv as special cases of a general mode of analysis. Even the simple problem of a circular cylinder of finite length, rotating about its axis, has not yet admitted an exact solution, though an approximate solution, which becomes valid when the cylinder is of infinite length, has been given by Chree. When the cylinder has a finite length, the surface con- dition of zero traction over the curved surface is violated, and instead of this traction becoming zero at all points on the surface, only its average value over the surface is zero. The results for the case of an infinite cylindrical annulas do no.t appear to be on record, and they are interesting on account of their marked divergence from those which belong- to the complete disk. In the present paper we group together some of the simpler and more interesting solutions of problems of this type, including those of the infinite circular cylinder and the infinite cylindrical annulus. These specific pro- blems are solved to any degree of approximation and for a non- uniform distribution of density. The analysis is simpler than is usual, for it does not seem necessary to treat these comparatively simple problems as special cases of general theory, and it is desirable, at least in the interests of the engineer or physicist, that a fundamentally simpler treat- ment should be placed on record. It also seems possible that such solutions may be of interest with regard to scientific instruments of great precision, in which some portion of the apparatus is in rotation, or, on the larger scale, in problems of practical engineering. Although no * Communicated by the Author. On the Rotation of Slightly Elastic Bodies. 31 novelty attaches to some of the earlier results, it seems desirable to include them. The simplest problem of! this nature is, of course, that of the thin circular hoop rotating about its centre. When such a hoop of radius a and density p is spun round its centre with constant angular velocity w the value of T, the tension per unit length in the Loop, is well known. For an element ds of its length has an acceleration aco 2 inwards, and the resultant of the tensions at its ends is Tds/a per unit area inwards. Hence the equation of motion is Tds/a = paco 2 , giving T = pa 2 co 2 . If, however, the hoop is slightly elastic, and X the value of Young's modulus for the material of which the hoop is made, and r the radius of the hoop when in motion, the equation of motion of the stretched element ds becomes T/V = ?'a> 2 . pair. Applying Hooke's law to the stretched element, we have, T = X(r-a)/a. Hence eliminating T, prato 2 = X(r — a) /a. In practice X is always large, and if we may neglect 1/X and higher powers of 1/X the appropriate value of r/a, which differs from unity by a quantity of order 1/X, is 1 + pa 2 co 2 /X. The value of the tension to the same order is pa 2 co 2 . The effect of a rotation is therefore to increase the radius a of the hoop to a(l +//,), where fj, = pa 2 co 2 l\, a number depending on the density, the elasticity, and the radius of the hoop, and on the rate at which it is rotating. As regards the practical order or magnitude of pa 2 (o 2 /X the extension per unit length, we may take a steel wire for which X is about 2*12 xlO 12 dynes per square centimetre, and p is about 7*5. In order that Hooke's law may hold, the extension per unit length must not exceed 10~ 3 , roughly speaking. If the velocity of a point on the rim is in the neighbourhood of 1*9 x 10 4 cm. per second — which is ap- proximately the case in a twenty-foot flywheel making two hundred and fifty revolutions a minute — we find that the extension per unit length is about 7*9 XlO -4 , which comes within the limits of applicability of Hookers law, and that 32 Dr. Dorothy Wrinoh on the the actual increase in the radius is about a fifth of an inch. In this case the tension is about 1*6 x 10 9 . It is further evident that </ n \/p is the largest velocity if an extension of more than n per unit length is to be avoided. When the elastic limit for the material is known, this result can be used to give an upper limit to the velocity it is safe to use if risk of deformation of the hoop is to be avoided. We may now proceed to the problem of a thin rod rotating about one end with uniform angular velocity. Thin Mod Rotating about One End. Let a be the unstretched length of the rod, co the angular velocity of rotation about one end 0, p the density when it is unstretched, and X the value of Young's modulus for the material of which the rod is made. Let T be the tension in any section in the rod during the motion. Let the distances of the same particle at rest and in motion be x and x. The density of the moving element dx is podx^/dx and its acceleration towards is xco 2 . The equa- tion of motion of the element is therefore, ST = — p Bxq . xco 2 , where, by applying Hooke's law to the element originally of length dx and now of length dx, we have T = \(dx/dx -l). Hence, eliminating T, we obtain the equation, d 2 x/dx 2 = — p Q m 2 x / A . The solution must give the value x = a when x = a if a is the length of the rod when in motion. Accordingly it is x = asm(x V p (o 2 / '\) / sin (a Q ^p Q oo 2 j\). We may determine the value of a by means of the condition that the tension vanishes at the free end, which is given indifferently by x = a or x = a . Thus, « \/p G) 2 /\ = tan a \^p co 2 /\. The equation relating the two corresponding positions of a Rotation of Slightly Elastic Bodies. 33 typical element when at rest and when in motion and the original length of the rod is therefore x \'p co' 2 /\ = sin (d? y/p w 2 /\) / cos (a vW*>7\)- Neglecting the cube and higher powers of l/\, we may replace this by the simpler form, as = a'o + 'i'opo(o 2 (3a 2 — tf 2 )/6\ to the order l/\. To the same order, T = p co 2 a 2 [1— x 2 la 2 ] . The greatest extension is ^p a) 2 a 2 /\, and this occurs at the end about which the bar is rotating. The tension is also greatest at this point and takes there the value p co 2 a 2 . As an example of the actual magnitudes of the quantities in practical cases we may take a twenty feet steel bar, which, when rotating about one end two hundred and fifty or three hundred times a minute, increases in length about a tenth of. an inch. dotation of an Infinite Elastic Circular Cylinder about its Axis. Passing now to a simple problem in three dimensions, we take the case of an infinite elastic cylinder of circular section rotating about its axis. We may consider one of the circular sections of the cylinder and use polar coordinates. At any point (r, 0) let r l\ and T 2 be the transverse and radial ten- sions per unit length, and T 3 the axial tension. We shall consider the motion of the element of volume which when at rest is bounded by the surfaces (z,z + 8z), (r, r + 6>), (0, + 80). By the symmetry of the cylinder, the element when in motion will continue to be bounded by the surfaces (0, + S0) : and since the cylinder is of infinite length, the element will continue to be bounded by the surfaces (z, z + hz). Let p represent the radial dimension, so that p — r is the radial extension at any point. Let a be radius of the cylinder and a its density, when at rest ; let &> be the angular velocity of the cylinder about its axis, and X and /x the elastic constants for the material of which the cylinder is made. The element of volume which we are considering is a parallelepiped of sides dp, pd0, and dz. The forces on our element of volume consist of (1) transverse tensions each Phil. Mag. S. 6. Vol. 44. No. 259. July 1922. D 34 Dr. Dorothy Wrinch on the o£ magnitude T±dpdz — and these are equivalent, in the usual way, to a radial force towards the centre of magnitude Tidpdsds/p, where ds~pd0; (2) of radial tensions T 2 ds dz and T 2 ds dz + dp d(T 2 ds dz)jdp, towards and away from the centre, which together give a £ ° rCe dpd0dzd( P T 2 )/dp away from the centre ; and (3) of longitudinal tensions each Fig.l. s p Ut 2 6s) T,t p/ .6s. of magnitude T s dp ds, in opposite directions. The resultant force then is simply dp dz ds (d(pT 2 )jdr — Tj) away from the axis, and perpendicular to the axis of the cylinder. The acceleration of the element is poo 2 towards the centre and its mass is ardrdO, since we may, of course, treat the density as constant over the element of volume. We therefore have the equation of motion, d(pT 2 )/dp — T 1 = — arpar dr/dp. d dr or Rotation of Slightly Elastic Bodies. 35 By Young's law we can express T 1? T 2 , and T 3 in terms of the extensions of the cylinder, in the well-known equations, T^Xidp/dr-l) + (X + 2fjL)(p/r - 1), T, = (X + 2fiXdp/dr-l) + \(p/r-l) 9 T 3 = \(dp/dr + p/r-2). Putting these values of T x and T 2 in the above equation we obtain the result ,(X + 2,)g-l) + ^-l)] -SKS-- 1 )-^^- 1 )-^ Qa/r) rf^/Wr 8 + W) [( L V dp/dr)/(\ + 2/i) + {\p/r)/(X + !>)] [dp/dr-p/r] = -p<7G> 2 /(A, + 2/x). . (1) The value of //, or X, varies from about 8 x 10 8 grammes' weight per square centimetre for steel to about 4 X 10 8 grammes' weight for copper. The corresponding densities are about 7 and 9 respectively. Terms involving 0-/0+2/*) are therefore of a smaller order than those which involve coefficients of the form _ ,, _ \ 2/*/(\ + 2,a). Let us write co 2 a — q{\~\-2pb). Then putting we can obtain a value for p to any order of approximation which is required. Neglecting, first, all terms involving q 2 . we have the equations for 77, p = r + qrj l ; dpjdr =l + q drjjdr ; d 2 p/dr 2 = q dPrjJdr* ; q [l + qvi/r] d?n 1 /dr*.+ llr[l + q(2/idf fl ldr)l(\ + 2p) + i*>ViM/(\ + 2/*) + q (d Vl /dr- Vl /r)] = -qr(l + ? % /r) ; q dfyjdi* (1 + y i/x/r) + 1/r [1 + ? (2/t* d Vl [dr)j(\ + 2/,) + frni/r)/[\ + 2/a)] 5 (d Vl /dr- Vl /r) +qr [1 + ^ r] = ; and since we are neglecting terms in y 2 , the equation for 77! is /•-' d 2 rji/dr 2 + r drj 1 /dr—rj 1 = — r 3 , D2 36 Dr. Dorothy Wrinch on the giving a solution or* the form Vi^a^ + bjr-^r 2 and b 1 = 0, since t] y is not infinite at the axis. The boundary conditions determine the constants a : and b x ; for the radial traction T x is given by T ] = (X +2 ,)(J-1) + X(^1) = q [{\ + 2 f L)d Vl /dr + \r ]l lr]\ and the radial traction must vanish over the two boundary surfaces. Thus (X + 2yLt) dffifdr | r =a +Xrj 1 /'r '< r =a =0. The constant a x is therefore determined by the equation, (\ + /x)a 1 = (2X + 3,a)a-78, giving T 1 = ? [2(X + /A )a 1 ~r 2 /4(2X+,>)] = 7(2X + 3/z)(a 2 -r 2 )/4r 2 and 7?1 = ^(a 1 r — ?' 3 /8) = ^[(2X + 3 / ,)(a 2 + & 2 )/ l X+/.)->- 2 ]. 77i£ Effect of a Circular Hole in the Cylinder. If the cylinder at rest has two boundary surfaces r = a, r=b (b<a) the solution stands in the form, Vl = (ll r-^b l /r~ir\ and the conditions for zero radial traction on both the bounding surfaces yield (X + /x)a l — fjbbi/a 2 + (2X + 3/a) a 2 /S, (\ + fi)a l = /ib 1 /b' 2 i- (2\ + 3yu &-/8. These give a 1 = (2X-I- 3/x) {a 2 + & 2 )/M(X 1 /*), b\={2\ + 3fM)a 2 b 2 l${i. Thus T 1 =g[2(X + /A )a 1 -2 / ,/Vr 2 -(2X + 3^> 2 /4] = 7(2x + 3/x)(a 2 — ?•'-') (r 2 — b 2 )/-kr 2 . Rotation of Slightly Elastic Bodies. 37 And the radial extension is p — r = <j \_a x r -\-b l ji — ? ,3 /8] o The maximum radial traction occurs where (« 2 -'- 2 )(>' 2 -i 2 )/>' 2 is greatest, namely, at r 2 = ab. The greatest radial traction therefore occurs on the cylinder whose radius is the geo- metric mean of the radii of the bounding surfaces of the cylinder. Its value here is pco%2\ + -V)(a - by/MX + 2fi). The transverse tension is given by To = q [ [\ + 2/z) Vl jr + 2\ dih/dr] = 7 /4[(2X + 3/x)(a 2 + 6 2 ) + (2\ + 3^)a 2 6 2 /r 2 -(\ + 2^> 2 ] = (2X+3^)|[a 2 + 6 2 -a 2 & 2 /^-(\ + 2^)r 2 /(2\+3/ A )]. The maximum value of this tension occurs on the cylinder with radius r 2 given by r 2 = abi/{(2\ + 3fi)/(\ + 2fi)}, and there its value is 7l2X ^ 3 ^[a 2 H-6 2 -2a/W(^ + L>)/(2X + 3^)]. Finally, the tension parallel to the length of the cylinder is _ fa •2 \ + 3/x X-\- a, <r + 1 2 )-2, 2 ]. The longitudinal tension T 3 is therefore greatest on the inner boundary. Here its value is q\ r(2\ + 3yu,Vr+ (fi-2\)h i\ n2\ + 3fju)a*+(fjL~2\ )(Sl 4 L X + /x J' In the well-known case of a solid cylinder, the longitudinal tension has its maximum on the axis, and its value there is (2X-h3yLt)^ 2 X/4:(X+u). 38 Dr. Dorothy Wrinch on the The radial and the transverse tensions have also their maximum value on the axis, where they take the same value, po) 2 (2\ + 3/x)/4a 2 (A + 2/x). The severest traction the cylinder is called upon to with- stand is therefore po) 2 (2A, + 3yu-)/4a 2 (\+2yu), and it occurs radially and transversely on the axis of the cylinder. In the case of the cylindrical annulus, the facts are entirely different. The longitudinal tension reaches its maximum value pco 2 X[{2Xi 3^a 2 /(\ + fi)-fjib 2 /{\ + fi)]mx^2fi) on the inner boundary. The transverse tension, in general, reaches its maximum value po>*(2\ + 3/*) [a 2 + b 2 -2ab V(X + 2m[2X + 3/*]/4(A, + 2<a), on the cylinder r = r 2 . And the radial tension reaches its maximum value f)co 2 (2\ + 3p){a-b) 2 mx + 2n), on the cylinder r=r 1 i As the ratio X//jl varies, the place at which the maximum tension occurs varies. If the ratio b/a is sufficiently large, i. e. if , /2\ + 3/x ,, . /a^+2/x b\f r t ->a, ba>\/— £-, V X+2[jl — ' ' ~ V 2X + 3/x the maximum transverse tension will occur on the outer rim ; otherwise it will occur within the cylinder. Thus a sufficiently thin annulus will have its maximum transverse tension on the outer boundary. For different values of the ratio X/fi, the maximum traction will occur transversely or longitudinally, for a cylinder for which the ratio a/b is given. And further, for a given ratio X/jjl, the maximum traction will occur transversely or longitudinally. Higher Approximations for Expansion of a Solid Infinite Cylinder. The first approximation for the radial extension was of the form, p — r-t-q {a x r — r*/8) = r + qn x . Rotation of Slightly Elastic Bodies. 39 To obtain a second approximation we may put f/'-i + ^i/^fW''; and therefore dp/dr = 1 -f qdr}i/dr + qhirf^dr and neglect terms involving ^ 3 . Using, again, equation (1) we obtain, if terms in q* are again neglected, d 2 tj 2 /dr 2 + drji/rdr — 7J 2 /r 2 = — rj l9 or r 2 d 2 rj 2 /dr 2 4- rdrj 2 /dr — r) 2 = — fl^r 8 + r°/8. The solution is of the form, r j2 = a. 2 r—a l r z /8 + r> 1 192. The radial tension T x is given by q [(X + 2p) d Vl /dr + X^/r] 4- q 2 [(X + 2/*) <fcfe/<*r + X V2 /r'] . The condition of zero radial traction on the boundary therefore yields (X + 2p) drj^jdr + Kvi/r + q[(X + 2p) d<n 2 Jdr + X7j 2 /r] = at r— a, or since the part independent of q already vanishes at r=a, (X + 2/jl) di) 2 ldr + Xrjo/r — at r = a. Hence T 1 = 7 (2\ + 3^)(a 2 -r0/4 + q 2 [(2\ 4- Sf^aja 2 - ?' 2 )/8 - (3X + 5p) (a 4 - r 4 )/S] , and since a 1 = (2\ + 3/A)a 2 /(X + /*) T 1 =q(2X+3fi)(a 2 ^r 2 )^ + V 2 [(2X + 3/*)V(a 2 - r 2 )/8(X + //,)- (3X 4- 5/*)(a 4 -r 4 )/8],, and, finally. ??2 /r = « 1 /[(2^-H3 A 6)a 2 /(X-r^)-^7 8 - [(3X 4- 5/*K/0 + /*) - »'*] / 24 x 8 _ 2X4-3/ , /2X 4- 3/, \ _ 1 / 3X + 5|* , _ A ~ 8(xTm) V~X+>~ / 8.24 V X+/i j* 40 Dr. Dorothy Wrinch on the Our second approximation is therefore, ■2X + 3/JL XH-3, X + /JL ' 2 r2\4-3/x 2 /2\+jV 2 2 \ 24\ A, + ^ }]' Any higher degree of approximation can also be obtained. Our results stand in the form, and, in general, the form will be y s lr = d r 's + ia s r 2 . . . + s a s r 2s .... (2) We easily see that if (2) be the form for ij s /r, then ' and the condition of zero radial traction, requiring that (X + 2/a) drig+ijdr + \r) s+1 /r = on r=a, yields a g 2X + 3/A 9 2«« 3\4 5/x, 4 uas+1 ~ 2 74TT7 a , + 176 ~Y+JT a ' * • 6 .a s (s + l)\+(2s+l)fi 2s + 2s-f-2.2s + 4 X + ^ so that W-2.4V ^ + y" J -4.-6V X + A* / + 2s + 2.2s + 4V \+> / W When we have obtained a solution, p-r = q Vl + q2 V2 .. . +^ SJ it is therefore possible to obtain a solution to a higher degree Rotation of Slightly Elastic Bodies. 41 of approximation by remarking the relation between the co- efficients in rj s jr and rj s +i/r and putting instead of as in 7j s l r, „ I ,— V / (» + l)X + (2n + ] y 2 „ + o ,,, + A Rotation of an Infinite Circular Cylinder of Non-uniform Density. We may next deal with the case of a cylinder in which the density is a function of the distance from the axis. Treating the case of the solid cylinder, we may put for the density of the cylinder when at rest, N *=J(r)=$a n r», o and N may haye any value from zero (when the density is uniform) to infinity — in which case the series '%a n r n must be convergent. The equation to be solved for a first approximation to the value of 7] is as before, d 2 7]/dr 2 + 1/r drjjdr — njr 2 = — aw'-rj (\ + 2/jl) . The solution is evidently, V =Ar- - ^ 2a M r»+7(( W + 3) 2 ~l). Tlie fact that there must be zero radial traction on the boundary surface r = a, yields the condition, T 1 =.(\+2/a) d V /dr + \r)/r = Q on r = a, giving 2(X+/*)A S [(n+3)(X + 2f*)+X] a n a*+ 3 /(n+4)(n+2j X+2/*o =*r^ S [(fi+4)X+2(« + 3)/*] a„a^ 2 /(w + 4)(n f 2). K-t-lfi 42 Mr. Gr. H. Henderson on the Consequently^ the value o£ T is r-^ r -2[(w + 3)(\ + 2 A t)+\] (a n+2 -r- +2 )a n /in+4,)(n^ 2) and 2 P ~ r ~\ + 2fi 1ST V >-l XanK +2 (n + 4\4-2/i + 3yL6)/2(\ + ^)-r"+ 2 }/(v^ + 2)(// + 4). Higher approximations to the value of rj can be obtained by the method adopted in the case of uniform density. IV. The Straggling of a Particles by Matter. By Gr. H. Henderson, M.A., 1851 Exhibition Scholar of Dalhousie University, Halifax , JW.S* § 1. Introductory . WHEN a parallel beam of « rays passes through matter, the particles gradually expend their energy in passing through the atoms of the matter, until all trace of the particles suddenly seems to vanish at the end of their range. In passing through the atoms some o£ the a particles lose more energy than others, so that at any point along their path some of the particles will be moving more slowly than others ; also their ranges will not all be the same. The a particles may be said to be straggled out, and hence the term straggling has been applied to "this phenomenon by Darwin. The theory of: the passage of matter by a rays has been developed on the basis of the nuclear structure of the atoms of the matter, and from this theory the amount of straggling to be expected has been deduced from probability, considera- tions. On the other hand, the straggling can be determined from experimental data in two ways. The first method makes use of ionization data. When the ionization due to a parallel beam of a, rays is measured at different points along the path of the rays, the well-known ionization curve is obtained. This curve is shown as the * Communicated by Prof. Sir E. Rutherford, F.R.S. Straggling of a Particles by Matter. 415 lull curve o£ fig. 1, where ionization is plotted as ordinate and distance from the radioactive source as abscissa. Now it has been shown experimentally that the (average) velocity of the a particles at any point of their path is proportional to the cube root of their remaining range. Assuming that the ionization produced is proportional to the energy lost by the a particle at any point of its path, it can at once be shown that the ionization should be inversely proportional to the cube root of the remaining range. Such a theoretical ionization curve is shown as the dotted curve of fig. 1. It Fis-. 1. Rcinge will be seen to be in approximate agreement with experiment over the first portion of the path of the a particle, but as the maximum is approached this agreement fails. G-eiger * has suggested that the ionization curve observed for a beam of a rays should be different from that of a single a particle, owing to slight variations in the ranges of the latter, i. e. to straggling. The ionization curve, built up of a large number of theoretical curves grouped around one ot average range, will thus be modified considerably near the maximum where the ionization is changing rapidly Hence the shape of the ionization curve near the end of the range should give an indication of the amount of straggling. * Geiger, Proc. Roy. Soc. A. lxxxiii. p. 505 (1910). 44 Mr. Gr. EL. Henderson on the Secondly, a more direct measure of the amount o£ straggling can be determined by counting the number of u particles at different points along the patli of a parallel beam. It is proposed in this paper to discuss the theoretical and experimental data on straggling, and it will be shown that the observed amount of straggling is much in excess of that allowed by theory. Further experimental evidence bearing on straggling will also be brought forward. § 2. The Straggling in Air. It might be thought that the individual a particles are emitted with slightly different velocities, thus giving rise to straggling. It has been shown by Geiger (loc. cit.), however, that the a particles emitted from a thin layer of radioactive material do not differ by as much as -J per cent, in initial velocity. Thus the cause of the straggling must be looked for in the air itself. As the a particle passes through the air it gives up its energy to the electrons and nuclei of the air atoms, and it is occasionally deflected through a considerable angle by close encounters with the nuclei. Different a. particles will encounter different numbers and distributions of electrons and nuclei and accordingly are straggled out. The calcu- lation of the consequent probability variations in the ranges of the individual a. particles has been carried out by both Bohr * and Flamm |- They agree in showing that the nuclei produce practically no straggling. They also agree closely in the amount of straggling produced by the electrons. The straggling of various types of a rays in air, calculated by Flamm's method, is shown in the second column of Table I. The value tabulated is the distance, measured along the range, over which the number of particles in a parallel beam falls off from '92 to 'OS of the original number. This corresponds approximately to the method of measuring the straggling from the experimental curves. The ionization curves for three types of a. rays have recently been determined with some accuracy by the writer %. The full curve given in fig. 1 is a reproduction of the ionization curve found for RaC. It was shown that the ionization curve from C to B (fig. 1) could be represented very approximately by a straight line. The slope of this straight line furnishes information as to the magnitude of * Bohr, Phil. Mag. xxx. p. 581 (1915). + Flarnm, Wien. Ber. II a. cxxii. p. 1393 (1913). X Henderson, Phil. Mag. xlii. p. 538 (1921). Straggling of a Particles by Matter. 45 the straggling. The. easiest way of considering the matter is to imagine the straight line produced in both directions till it meets the axes or! zero and maximum ionization at D and E. Then the projection of the line DE on the axis of zero ionization (or the reciprocal of its slope) is a direct measure of the straggling. The greater the straoro-lino- the greater will be the projection referred to, and as a first approximation the projection may be taken as proportional to the straggling. The values of the projections taken from the writer's curves are given in the third column of Table I. That for polonium has been determined from the curves given by Lawson *. The curves obtained by counting the number of a particles in a parallel beam at various points along the path show that this number remains constant till near the end of the range and then falls off rapidly to zero. Most of this falling off also approximately follows a straight line. The reciprocal slope or projection of this line is a more direct measure of the straggling than the corresponding projection of the ionization curve. Measurement of the straggling by means of counting experiments is, however, very slow, as large numbers of a particles must be counted. The values obtained from the most recent and reliable counting experi- ments are given in the fourth column of Table I. The result for polonium is taken from a scintillation curve given by Rothensteiner f ; that for RaC is from a curve obtained by Makower { by photographic counting of the a particles. All the results in the Table are in millimetres and refer to air at 0° C. and 760 mm. Table I. Type of Kays. Theoretical Straggling. Straggling from Ionization Curves. Straggling from Counting Expts. •88 102 144 1-74 3T 40 ; Thorium G x 2-88 2-83 2-92 4T : Thorium C 2 This Table shows clearly that the observed values of the * Lawson, Wien. Her. II a. cxxiv. p. 637 (1,915). t Rothensteiner, Wien. Ber. II a. cxxv. p. ]237 (1915). % Makower, Phil. Mag. xxxii. p. 222 (1916). 46 Mr. Gr. H. Henderson on the straggling are three or four times greater than those predicted by theory. Furthermore the calculated straggling increases steadily with increase of range, while that observed is constant within the limit of error. It should be pointed out that the projections given in the Table are measured as the small differences between two larger quantities, and hence are more difficult to determine with accuracy. The straggling deduced from the writer's experiments has a probable error of about 2 per cent., and it will be seen that the values for the three types of rays agree within this limit. It was shown by the writer (loc. cit.) that the effect of straggling due to electronic encounters would be a tailing off' of the ionization curve at the extreme end of the range. Making some simple assumptions it w r as shown that the calculated form of the end of the ionization curve agreed satisfactorily with the form of the curve observed between A and B (fig. 1). Thus the effect of the calculated straggling- was amply accounted for by AB, leaving the much greater straggling evidenced by the straight line portion BC quite unexplained. The curves obtained by counting experiments also lead to precisely the same conclusion. In view of the failure of theory to account for this large excess straggling- it is interesting to see what further information regarding it can be derived from experiment. It is remarkable that the straggling (as measured by the projections of the ionization and also of the counting curves) should be constant for a rays differing so widely in range as those given in Table I. This can only mean that the excess straggling takes place only in the last tw r o or three centi- metres of the range. From experiments with gold foils which will be discussed later, it appears probable that the straggling is confined to the last few millimetres of the range. Referring once more to fig. 1, it could not be expected that the straggling deduced from the ionization and the counting data would agree, for the following- reasons : — The ionization curve is regarded as being built up of simple curves of different ranges grouped about a common mean. The form of the simple curve is not accurately known ; the rule that the ionization is inversely proportional to the cube root of the remaining range can only be an approximation to a much more complicated law. As the shape of the simple curve cannot be taken into account, the projection of the ionization curve which is actually utilized can only give a . Straggling of a. Particles by Matter. 47 rough indication of the absolute magnitude of the straggling. However, comparative values of the straggling under different conditions should be given fairly accurately by the method adopted. On the other hand, in the counting experiments the assumption is made that the zinc sulphide screens or photographic plates used have the same efficiency for a rays of low speeds as for those of high speed. This assumption is not altogether justifiable. § 3. Straggling in Gases other than Air. The ionization curve in hydrogen was determined with the same apparatus already used for air. The gas was obtained from a cylinder of compressed hydrogen stated by the makers to be of more than 98 per cent, purity. Small impurities are unlikely to affect the straggling- mate rially. It was again found that a considerable portion of the eud of the ionization curve could be represented by a straight line. When the range of the a. particles was reduced so as to give the same range as in air, the projection of the straight line was 2*05 mm. with a probable error of 3 per cent. The straggling in air and hydrogen may be deduced from the ionization curves given by other observers. The results agree in every case within the limits of error, although the conditions for accuracy were less favourable than in the present experiments. The value 2'0 mm. for polonium in hydrogen may be obtained from some ionization curves given by Taylor *. From the results of Lawson (loc. cit.) for polonium the straggling was determined as 3"1 mm. in air and 2'2 mm. in hydrogen. The straggling in oxygen has also been deduced from experiments made in that gas with the present apparatus, using ThC. The value found was 3*36 mm., when the range was increased to the same value as in air. The straggling in several other gases may be deduced from the ionization curves given by Taylor (loc. cit.), although the error involved is probably of the order of 10 per cent. The collected results of stragglingin gases are given in fig. 2, which shows the straggling plotted against molecular stopping power. The values plotted for air, hydrogen, and oxygen are from the writer's results; the remainder are taken from es. * Taylor, Phil. Mag. xxi. p. 571 (1911). 48 Mr. G. H. Henderson on the Fig. 2 can only be considered to give an approximate idea of the .facts, as the points are not well distributed and some may be seriously in error. It would seem, however, that the straggling increases very slowly as the stopping power of the gas is increased. It is unfortunate that the dearth of suitable gases of high stopping power makes the checking of this point difficult. Fig. 2. . 4 £ E c 3 O") c o-» 2 <3 if) t so, £ H * a O 2 J*-*—"" Air *'" cs d <^ H ,o° . ._!_. _JL_ ... .1- 1 a Molecular S+o pp.ng 3 Power. § 4. T7i£ Straggling due to Solids. The great difficulty which at once arises in determining the straggling due to solids is the uneven thickness of the solid foils used, the effect of which may completely mask the true straggling looked for. An attempt to avoid this difficulty was made by using a large number of the thinnest beaten foils of the solid obtainable ; with gold, for example, as many as 128 thicknesses were used. Composite sheets of gold and other metals were placed immediately over the radioactive source (ThC) and the ionization curves deter- mined in air with the same apparatus as before. Although a rough calculation seemed to show that the irregularities in the individual foils would be smoothed out enough to avoid masking any true increase in straggling, this result was not borne out by experiment. It was finally concluded that the increase in straggling observed was mainly, if not entirely, due to unevenness of the foils, and hence need not be gone into in detail here. In mica the increase in straggling was much the smallest, as was indeed to be expected. Fortunately, experiments on the straggling produced by Straggling of a. Particles by Matter. 49 solid foils for low velocities of the a particles gave results which were not masked by irregularities of the foils. In these experiments the foils were placed 3 mm. from the middle of the ionization chamber (itself 1 mm. dee])) in air at a pressure of roughly 17 cm. Reduced to air at normal pressure the distance from foil to centre of ionization chamber was therefore about "7 mm. Most of the foils used were made up of a few T thicknesses of gold leaf. The air equivalent of a single sheet was about *I5 mm. when placed directly over the source ; when placed near the ionization chamber the air equivalent was about '28 mm. The straggling of the a particles after passing through these foils was deter- mined in the same manner as before from the ionization curve in air. The results are given in Table II. The straggling is in millimetres, and the probable error is about 2 per cent. The third column shows the straggling observed when the foils are placed directly over the source, the steady increase with increasing number of leaves being mainly due to unevenness of foils. The fourth column shows the increase in straggling at low velocity over that at high. This .increased straggling is real and almost independent of the unevenness of the foils. Results with aluminium (1*0 mm. air equivalent) and mica (S'6 mm. air eq.) are also included in the Table, but with these foils the change observed is scarcely more than the experimental error. Table II. No. of Leaves ( ^™ 8 loHifa«on Stra ^ in S Foil 111 FolL Chamber. near Source ' Difference. 2-88 1 3-30 4 ■ 3-65 12 400 2-88 2-92 301 3-28 3-01 306 •38 •64 •72 •07 •10 Aluminium ' 308 Mica 3-16 These results show quite clearly the rapid increase of straggling near the end of the range. One gold leaf nearest the ionization chamber causes nearly twice as much straggling as three leaves immediately behind it. Eight more leaves Phil. Mag. S. 6. Vol. 44. No. 259. July 1922. E 50 Mr. G. H. Henderson on the placed behind these four again only slightly increases the straggling. The same result was also demonstrated in a slightly different manner. A foil made up of the four gold leaves already used was placed at different distances from the ionization chamber and the straggling determined from the ionization curves as before. The results are shown in Table III. The distances given in the first column are not the actual distances from the foil to the centre of the ionization chamber, but are reduced to correspond with an ionization vessel containing air at atmospheric pressure. Table III. Distance from Ionization Chamber. Straggling. •7 mm. 4-7 „ 12-9 „ 3'65 mm. 3-26 „ 2'95 „ Here again it can be seen that the straggling increases rapidly near the end of the range. It may also be noted that the straggling (2*95 mm.) when the foil was 12*9 mm. from the ionization chamber is less than that (3 01 mm.) obtained when the foil was placed directly over the source. The difference is less than the possible error, but such a change is to be expected. For the straggling due to unevenness of the foil should be less for low velocities of the a particles on account of the decrease in air equivalent of the foil. For the same reason the differ- ences given in the fourth column of Table II. are probably slightly too small. From the straggling observed with different types of ol particles it was pointed out in § 2 that the straggling must occur within the last two or three centimetres of the range ; from the results with gold foils it seems that the straggling must be confined largely to the last few millimetres of the range. The increase of straggling due to gold foils placed near the end of the range, though clearly marked, is small. This is quite in accordance with the view expressed in § 3 that the straggling does not increase rapidly with increase o£ stopping power of the substance causing the straggling. It must be remembered that the increase - observed is the Straggling of u Particles by Matter, 51 increase over the straggling which would be produced by a layer of air equivalent to the gold foil used. The ionization must be measured in a gas such as air at a reasonable pressure, and hence we have the complication of the straggling due to the air between the solid and the ionization chamber, and even in the chamber itself. This sets a limitation to the amount of information to be obtained From ionization data. Accordingly the ionization experiments were not carried beyond the stage described, but it is hoped to push further the attack on the problem by more suitable methods. An isolated experiment with iron foils may be referred to before closing this section. A sheet of iron of about 4 cm. air equivalent was placed directly over the source and the ionization curve measured when the iron was magnetized parallel with and perpendicular to the direction of travel of the a particles. The purpose of this experiment was to see if there was any change in the straggling due possibly to rearrangement or change of orientation of the electrons in the iron, a point which has been discussed by Flamm *. Alternate readings of the ionization with parallel and per- pendicular fields of about 100 gauss were made at various points along the range. No appreciable difference could be detected- It should be added that the iron was very uneven in thickness, and the consequent; straggling was so large (about 20 mm.) that a small change in straggling might well have been masked. § 5. Summary and Conclusion. It has been shown in this paper that the straggling of « particles, as deduced from both ionization and counting experiments, is several times greater than that deduced from theory based on our present views of the mechanism involved in the passage of a particles through matter. It has been shown in a previous number of this Magazine that the effect of the calculated straggling can be adequately accounted for by the tailing off of the ionization curve at the extreme end of the range. The large additional straggling observed behaves quite differently. Evidence has been given in this paper to show that it increases very slowly with increase of molecular stopping power, and furthermore, that it all takes place within the last few millimetres of the range. Here we seem to be confronted with a behaviour of the a part cle which present theory is unable to explain. * Flamm, Wien. Ber. II a. cxxiv. p. 597 (1915). E 2 52 Dr. G. Green on Fluid Motion Evidence obtained from the Wilson photographs also leads to the same idea, for it has been shown by Shimizu * that the observed number of ray tracks which break up into two branches near the end of the range is much greater than the number deduced from probability considerations based on our present theory of atomic structure. It is noteworthy that this anomalous behaviour of the a particle occurs at low velocities, where practically no investigation of the scattering of a particles has been carried out on account of the experimental difficulties of dealing with slow ol particles. It is at higher velocities, where the theory of scattering put forward by Sir Ernest Rutherford has been so fully verified by experiment, that the most of the theoretical straggling takes place, and this straggling- has apparently been accounted for. In conclusion I wish to express my best thands to Professor Sir Ernest Rutherford for his kind interest and advice. I also wish to thank Mr. Crowe for the preparation of the radioactive sources. Y. On Fluid Motion relative to a Rotating Earth. By George Green, F>.Sc, Lecturer in Natmal Philosophy in the University of Glasgow f. THE subject of this paper is at present one of consider- able interest to meteorologists. Papers by the late Dr. Aitken and also by the late Lord Rayleigh on the dynamics of cyclones and anticyclones have been followed by more recent papers by Dr. Jeffreys, Sir Napier Shaw, and others. Very few actual solutions of the equations defining atmospheric motions have been obtained. In the late Lord Rayleigh's paper % attention is drawn to certain general hvdrodynamical principles relating to the properties of rotating fluid which can be applied to " assist our judg- ment when an exact analysis seems impracticable." The importance of the theorem regarding the circulation of the fluid in any closed circuit is clearly explained in its applica- tion to any actual fluid motion. In applying this theorem to fluid motion in the atmosphere, however, we must bear in mind that the motions with which w r e are concerned are not the actual motions of the particles in space but their motions relative to the Earth itself at each point of observation. * Shimizu, Proc. Roy. Soc. xcix. p. 432 (1921). f Communicated by the Author. X Sc. Papers, vol. vi. p. 447. relative to a Rotating Earth, 53 One object o£ the present paper is to investigate the con- ditions under which the circulation theorem may be applied to atmospheric motions relative to the Earth's surface ; or more generally to motions relative to any three rectangular axes which are themselves rotating about each other, with a fixed origin. In the later part of the paper one or two additional cases of motion of the atmosphere are discussed and the system of isobars corresponding to each motion determined. In view of the problems to be considered, we shall begin by specifying the system of rotating axes most convenient in dealing with fluid motion in the neighbourhood of any point of reference on the Earth's surface. The axis OZ is drawn upwards along the apparent vertical at 0, and line OZ continued downwards meets the axis of the Earth at a point 0' which is taken as origin of coordinates. Then axes O'X and O'Y are drawn parallel to horizontal lines through the reference point in directions due East and due North respectively. In the most general case to be considered the reference point may be in motion relative to the Earth's surface, and this involves also a motion of the origin 0' if point moves either North or South. But the motion of 0' corresponding to any moderate motion of is very small, and for our present purpose we may regard the origin 0' as a fixed point, very near to the centre of the Earth. We shall denote by (.r, y, z') the coordinates of any point referred to origin 0', and by (#, ?/, z) the co- ordinates of the same point referred to parallel axes through 0. This makes z r = z + H, where R represents approximately the radius of the Earth. The components of the velocity of any particle relative to the axes at any instant are repre- sented by u, r, w, and the angular velocities of the axes themselves, that is, of each two axes about the third, are represented by ft> z , ay^ a) z , respectively. We shall introduce the particular values of ao x , w y , co z corresponding to a reference point fixed in position on the Earth, or moving relative to the Earth, when we come to deal with special problems. Referred to the above system of axes, the equations of motion of any fluid particle take the form : — Dz < , a - <* V 1 "dp ( i n 54 Dr. G. Green on Fluid Motion In these equations, 1? 2 , Z represent the terras depending on the rotation of the axes themselves, being given by equations of the type #i = — 2o) z v + 2<Djiv — w : y + Wy.z 1 + co x (o y i/ -f ov^s' — (co/ + &)/) 0. ... (4) The function V(#, y, *) represents the gravitational potential function. We have also D 3 ^ 9 j. „ ^ j. * «•> D7 = 5t + l( ^ + ^ + "5i- • • • (5) The equation of continuity of the fluid is then W + K^ + ^ + 9l)=°- • • • (b) In applying these equations we treat the atmosphere as a perfect gas in which viscosity may be neglected. Circulation Theorem for Relative Motion. Consider now the theorem relating to the relative circula- tion. We have j^ (u dx + vdy + w dz) = y- dx + jydy + -^ dz + d (i q 2 ), • • • (?) where q 2 = u 2 + v 2 + w 2 , the square of the resultant relative velocity. By means of equations (1), (2), (3), the above equation may be rewritten in the form : yr- (u dx + v dy + w dz) = -(0 L dx + 6 2 dy + s dz)- C ^-dY + d(iq 2 ). (8) We can now integrate each term of this equation along any curve within the fluid from any point A to any point B. This integration gives the result, Dl'B rs C dp n I (udx + vdy + wdz) =—] (#i dx -\-0 2 dy + O s dz) — 1 - J - -V B + V A + i?B 2 --kA 2 ; (9) and, if the integrations are applied to a closed curve relative to a Rotating Earth. 55 beginning and ending at the point A, we obtain D C C jj-l (uds+vdy+wdz)= — \ (d l dx + e 2 dy+6 z dz), (10) v s * s where the suffix S indicates that the integration is to be taken along a definite curve S. We have assumed in obtaining (10) from (9) that V is a single valued function of («r, i/, z), and that p is a f miction of p. It now appears that the rate of change of the relative circulation in any closed circuit which consists of the same fluid particles at all times is not zero unless, in addition to the above con- ditions, we have B01_d03 ^2_Bfl ^3_^2 nn When these conditions are not fulfilled, the relative vorticity does not move with the fluid itself, and if a velocity potential exists for a certain portion of fluid at a given instant, a velocity potential will not exist for that portion of fluid at a later instant. The first case of importance of the above conditions in relation to problems relating to the atmosphere is that in which the angular velocities co x , o> y , co z of the axes are constants. In this case, the conditions given above take the form 1 * 1 o 1 £ "bit "dv "dw n9 N (D r Wy co, o<v oy 0% where 8 represents an operator defined by '-(•4, + +fr-h)- • • • (13) These equations have a solution of the form u v ic ,-^= «=/(•*+-#+-/)> ■ ' (14) where /denotes any arbitrary function. If we draw an axis to coincide with the axis defined by the resultant of the three component rotations co X) co y , co z , then [co x x + co y y + co z z) is equal to URcos^), where fl is the resultant of (a> x , a> y , (*> z ) and R is the line joining the origin to the point a, y, z. That is, u, v, w are functions of p the perpendicular from the origin to a plane through the point (<r, ?/, z) perpen- dicular to the axis of the resultant rotation. 56 Dr. Gr. Green on Fluid Motion When the fluid is incompressible, and when a compressible fluid is moving m such a way that ^ — h ^ — \- %- is zero, B# oy d~ a solution of a different type obtains. The solution in this case may be written in the form u I =/i{(V-^), (a z x-co x z)}, v=j 2 {{a) 1J x-G) x y), (ay z x-co x z)}A. . . (15) w =/ 3 { (aye - <o x y) , {co z x - m x z) }, ) where f 1} f 2 , f% are arbitrary functions subject only to the condition — + ~- + ^- =0. This solution includes as a ooc dy oz particular case any motion of rotation of the atmosphere as a solid about the axis of the Earth. The solutions which we have above obtained make it clear that the fluid motions relative to rotating axes in which the relative circulation moves with the fluid belong to a very restricted type. A relative motion, for instance, similar to that taking place in a free vortex, does not fulfil the con- ditions required for permanence of the velocity potential, and therefore no steady motion of this type could take place in the atmosphere — as has been assumed to be the case. The conditions which we have found to be necessary for the validity of the circulation theorem when the fluid motion is relative to rotating axes, may be obtained in a manner different from that employed above. Taking H, H, Z to denote the components of angular velocity of a fluid ele- ment, and U, V, W to denote components of linear velocity of the element, each referred to fixed axes which coincide at instant t with the instantaneous positions of the moving axes, we may derive the conditions from the equations employed by von Helmholtz in his papers on vortex motion : — with two other similar equations. With £, rj, £ to represent the components of relative angular velocity of an element of fluid, referred to the moving axes, we have, JJ = u — (o z y + cOyZ; Y = v — co x z + a) z x; W = to — w y x + w x y ; and Dt l)t ~ a ' v W ^' relative to a Rotating Earth. 57 By means of these relations we can readily transform (16) and obtain the corresponding equations for the rates of change of the circulation components of an element of fluid referred to the rotating axes ; in this way we find Df <-B« . B« . V B" f./'d" , Bw . "dw\ d# B// Z B^ \B.i' By Bs/ with the corresponding equations in rj and f. Now the hydrodynamical theorem that relates to the permanence of a velocity potential for the motion of a given portion of fluid and the theorem of the permanence of the circulation of an element of fluid depend on equations (16). The equations which we have obtained for the relative circula- tions reduce to these equations exactly when the conditions expressed in (12) are fulfilled ; and these conditions must accordingly be fulfilled in order that the theorems referred to may apply to the relative motion, in the same way as they apply to the actual motion. Particular Cases of Motion Relative to the Earth, We shall now discuss one or two particular cases of fluid motion relative to the Earth, and we shall, to begin with, take the reference point O as a fixed point on the surface of the Earth at latitude (j> degrees North. The angular velocity components w x , a> v , co z have then the values 0, fl cos <£, 12 sin </>, respectively, where fl represents the angular velocity of rotation of the Earth about its axis. If we now let zr represent the perpendicular distance from any point #, y 9 z to the axis of the Earth, we can write the equations of motion (1), (2), (3), in the form where V'=Y— JDV. V is, in fact, the potential function dV gx 3V gy av a« ~ r ' ay R' S* 58 Dr. G. Green on Fluid Motion TsY' corresponding to apparent gravity, so that — ^ — at point (# = 0, y = 0, £ = 0) is the value of —g at the reference point 0. In applying the above equations to motion of the atmosphere, we may take av av av t . in the immediate neighbourhood of in the region within which the value of apparent gravity may be regarded as constant in direction and amount. If we neglect a change o£ direction of one degree in g, our equations (18) to (21) would then represent conditions of motion within a radius of about seventy miles from point O. In order to render our equations suitable to represent circulations of air of diameter exceeding, say, 150 miles, we might employ the approximate values , . . (22) wherein we neglect the variation of g with height. If we exclude certain cases of motion relating specially to the tides, very few solutions of the above equations have been recorded. In order to make ourselves familiar with the types of fluid motion possible in the atmosphere it is of interest to examine all solutions which can be obtained having a bearing on meteorological problems. We, accord- ingly, take first the steady rotational motions of incom- pressible fluid under the force of gravity alone. We may take the boundary condition w = Q to apply at the surface of the Earth. A simple rotation of all the fluid about a vertical axis through 0, with a uniform angular velocity o>, would be represented by u=—(oy; v=+(Die; io = 0. . . (23) With these values in equations (18) to (20), it would be impossible to satisfy (19) and (20) simultaneously ; but the motion represented by u= — co(y — /3z) ; v = cox\ io = 0, . . (24) fulfils all the conditions contained in our equations, pro- vided (3 is given by o_ 212 cos j> m . relative to a Rotating Earth. 59 and the pressure p is given by ? = - V + i(a> 2 + 2a>n sin (/>) {.i< 2 + <j/-/3c) 2 } + & . (26) » r The approximate value of V' is +g[ 9T / + - ) , and p is the pressure at the reference point 0. In the motion indi- cated above each particle of fluid moves in a horizontal circle whose centre lies in the line (y=/3z, a? = 0). This line lies in the meridian plane through point and is inclined to the vertical at at an angle 6 towards the North, where tan<9 = /3. When a) is very small in relation to H this in- clined axis is almost parallel to the axis of the Earth ; and when co is large this axis comes almost to coincidence with the apparent vertical at 0. With co very large the motion described above corresponds very closely with a uniform rotation of the fluid about a vertical axis, as in the case of a simple forced vortex. Another case of motion of incompressible fluid of interest in the same connexion is that represented by .,, u=—coy; v = cox; w = 2X2 cos 6 . a?, . (27) with T J - = -V' + i (co 2 + 2vn sin (f>){% 2 +y 2 ) +2W cos 2 cb . x 2 + & , . . . (28) as the equation showing the distribution of pressure. This motion differs from that first discussed in not being exactly horizontal. The plane of motion of each particle of fluid passes through the line OX, and is inclined to the horizontal plane XOY at an angle 6 given by tan = 2Q cos <p/co. When the angular velocity of rotation co is very large com- pared with flcos</), the plane of motion of each particle is practically horizontal, and the motion then corresponds very closely with that of simple rotation of all the fluid as a solid about, a vertical axis. When co becomes small, on the other hand, the inclination of the plane of motion of each particle of fluid to the horizontal increases. The two motions, repre- sented by (24) and (27) respectively, are almost identical when ft) is very large, and they differ entirely when co is very small. It would be interesting to investigate the manner in which a fluid, such as water, subsides to rest from an initial condition of steady rotation about a vertical axis. The solution represented by (24), (26), would appear to be the exact solution for steady rotation of water in small scale experiments. 60 Dr. Gr. Green on Fluid Motion The motions considered above are motions of any incom- pressible fluid and do not indicate, except as approximations, conditions of motion possible in the atmosphere, One or two solutions of a similar type can be obtained which refer to incompressible fluid and accordingly represent motions possible in the atmosphere. Consider now the motion repre- sented by «=-©(y-^); 5 v=0; m? = 0. . . (29) The equations of motion of any element of fluid in this case are °=- 9 i- k h logp ' ■ (30) -20 sin </> . oo(y-!3z) = ~ || -k^-logp, . (31) 2OooB*«(y-^)=-^-^lo g/0 , . (32) and these equations are satisfied provided the pressure system throughout the fluid is that indicated by ilogp = n«a*. a>(y-/3zy- 9 (^f + «) + Mogp , (33) where /3 = cot.$, and p is the density of the fluid at the reference point O. The continuity equation is also satisfied provided we can neglect the term (go)/R),v(y — ftz). This condition limits considerably the extent of the region around to which our solution is applicable, as stated earlier. Within the region to which the above applies the isobars at the surface of the Earth run East and West, being determined by h log p = 12 sin </> . coy 2 -f k log p Q , . * . (34) which indicates a system symmetrical on the two sides of the East and West line drawn through reference point 0. In this case the isobars become closer as we proceed North or South from point 0. They are also parallel to the lines of flow of the fluid. The coefficient ^ has the value 1*5 x 10~ 6 with the foot and the second as units; while 2D sin (j> has the value 1*03 x 10 ~ 4 at latitude 45°. Certain cases of interest arise in which the terms containing ~ may be neglected. For relative to a Rotating Earth, 61 example, we may take the case of a uniform east or west wind over a considerable region. In this case u = c, r = 0, w = 0, . . . . (35) and the pressure distribution consistent with this motion is represented by k log /3 = 2 Or (cos </> z — sin (j>y) — gz + k\ogp , . (36) where p Q refers to the density of air at the reference point 0. The isobars at the Earth's surface are in this case a uniform system running due East and West. A similar case is that given by u=0, v = cx + d, 20 = 0, . . . (37) which represents a wind towards the North, while the corre- sponding pressure distribution is that represented by k log p = O sin (/> (ex 2 + 2dx) —gz + h log p . . (38) The isobars are again a system of straight lines, but runnino- north and south, and uniformly spaced when c = 0. In a similar manner we find that u = c 1 , v = c 2 , iu = 0, . . . . (39) corresponds to a system of straight isobars represented by k \ogp = 2Q sin cf) {c^x — c^y) + 2Q cos<f> .c^—gz + k log/j . (40) The isobars are again lines of flow of the air, as in each case considered above. The case of motion corresponding most closely to a cyclonic or anticyclonic circulation is that discussed in an earlier paper *, represented by it=—<o(y-(3z); v = co l i-; iv = 0. . . (41) In this case Q _ 2fl cos <j) ^- w + 2min^,' < 42 ) and the pressure distribution is that represented by ^log^ = i(a) 2 4-2a>f2sin</,){.r 2 -r( i/ -/3c) 2 }--^- f ^log /0o . . . . (43) The term fr^- must be small in order that the continuity equation may be fulfilled. * Phil. Mag. vol. xli. April 1921 ; vol. xlii. July 1921. 62 Prof. G. N. Antonoff <m the In each case considered we have only to replace O by an increased value fl' to obtain a motion for which the system of isobars travels eastward at a uniform speed (CI' >Q). Each of: the above solutions has been given to apply to an isothermal atmosphere, and in every case considered the fluid moves so that no element of fluid undergoes change of density. That is, ~ — h s: — H ^— =0 at each point. Provided "* 3# dy d* r this condition is fulfilled, any solution obtained for motion of an atmosphere, all at one temperature, can readily be trans- formed to suit an atmosphere in convective equilibrium (p = Jepy), or one in which pressure is any given function of density. Thus, taking the motion represented by (41) above in an atmosphere in convective equilibrium, we have merely to replace log/o and log/o in (43) by — —. ,P y ~ l and - po 7 " 1 respectively, all other conditions being unchanged. VI. The Breaking Stress of Crystals of Rock- Salt. By Prof. G. N. Antonoff, Z>.Sc.(Manch.) *. IN a paper published in Phil. Mag. vol. xxxvi. Nov. 1918, I have developed a theory of surface tension under the assumption that the attraction of molecules is due to electrical or magnetic forces, or both. Instead of assuming a uniform field round the molecules as it is generally accepted acccording to Laplace, I accepted the view that the molecules act as electrical doublets, and from the theory of potential I deduced that the attraction between them must be inversely proportional to the 4th power of the distance, provided the distance between the doublels is large compared with their respective lengths. It was shown that the attraction between the doublets can be represented by an expression of the type kP where k is a constant, I the length of the doublet, and d the distance between them. In these calculations the magnetic forces were disregarded altogether, as the law of attraction between small magnets would be just the same, so that they could only have an effect on the value of k. * Communicated hy Dr. J. W. Nicholson, F.R.S. / d> Thus P 9 / S l-V 1/E Breaking Stress of Crystals of Rock- Salt. 63 For the surface tension, the expression was given as d or assuming ~r=P where p is the number of molecules per unit volume, the expression for the surface tension a becomes u = kl 2 p^; instead of p we may put l 2 , where 6\ is the density of the liquid, B 2 that of the saturated vapour, and M the molecular weight. -=«■(¥•)" (i > It was also shown that the internal pressure P can be calculated by the formula P = 2«p 1 / 3 (2) ) (3) In other words, the intrinsic pressure can be calculated from the surface tension if the molecular weight of the liquid is known. Thus for normal or non-associated liquids it should be possible to calculate the normal pressure from the value of the surface tension. It should be pointed out that the assumption made in the above theory, that the length of the doublet is small compared with the intramolecular distance, is not necessarily the right one. In the expression (2) this law is, however, eliminated, and the same expression is obtained for any law of molecular attraction. The figures for P obtained from the above expression agree with those from indirect evidence. How- ever, experimentally it is not possible to determine P directly, owing to the mobility of the particles of liquids which always adjust themselves so that the molecular pressure is inappreciable. It is not so in the solid or crystalline state, in which the particles have a definite orientation, and where the internal pressure can be determined by a direct experiment. It is sufficient to apply to a crystalline body such a weight as would overcome the attraction of the molecular forces and cause the disruption of the body. The force applied is not 64 Prof. G-. N. Antonoff on the necessarily the same in all directions, and it is therefore necessary to specify the direction in which it is to be applied. The question arises now whether it is possible to calculate the surface tension of a solid body. For the solid state there is no direct method of determining the surface tension, all methods used for liquids being inapplicable in this case. Some attempts were made to estimate the surface tension of solids from indirect evidence. For example, Ostwald * and Hulett f calculated the surface tension of some calcium and barium salts on a basis of a certain theory from the solubility data. The figure given for the latter is about 4000 dynes per cm. From the point of view of our theory, it seems possible to calculate the surface tension by the use of formula (2) by determining experi- mentally the internal pressure per square cm. of the cross- section, if the molecular structure of the substance in the crystalline state is known. At the present time the X-ray analysis throws a light on the above question. • For example, according to W. H. and W. L. Bragg, the crystal of rock-salt consists of charged ions situated at regular distances from one another. Such a case is somewhat different from the one discussed in my paper {loc. cit.). Here it is necessary to assume that l = d, where / is the length of the doublet and d the distance between them, under which conditions the ordinary inverse square law must hold true. The attraction between the charges in a row is equal to where e is the elementary charge, and the value of k is the sum of a series 1 — i + J — i + ^— • • • =0*6931. Assuming that the adjacent rows have no effect upon the charges, the expression for the surface tension is of the form where p — number of particles per unit volume. For the normal pressure the expression will be * Zeit. Phtjs. Chem. xxxiv. p. 503 (1900). t Zeit. Phys. Chem. xxxvii. p. 385 (1901). Breaking Stress of Crystals of Rock-Salt. 65 For rock-salt the density = 2*15, M= 1*64 x 58-5 xlO- 28 , p= ~ =2-24 xlO 22 ., Assuming that k is approximately = 0*7, the tension is e 1 x 0*7 x 2-24 x 10 22 = 22'1 x 10" 20 X 0-7 x 224 x 10 22 = 3500 dynes per cm. (approximately). The figure obtained is of the same .order of magnitude as figures derived by Ostwald and Huletfc (loc. cit.) for barium salts. The normal pressure P would be accordingly P = 98"7 X 10 9 dynes per square cm. It is interesting to see now how far the above results agree with the experimental evidence. An experiment was performed as follows : — I took a good specimen of rock-salt crystal and I cut a prism of the section about 15 square mm. and about 2-3 cm. long. I used a suitable cement to hold the piece from both ends, and by applying a suitable weight produced a rupture of the crystal into two halves. Measuring the cross- section of the rupture accurately, I calculated the weight required to produce the rupture per square cm. I have repeated the experiment many times with different samples of rock-salt. If the crystal is well formed, the agreement between individual experiments is fairly good. In one series of experiments, I cut the prisms so as to have several samples cut parallel to the three principal axes. I have done the experiment with three pieces for each direction. For one direction I obtained : 89 lb. per cm. 103 „ „ 83 „ „ In the other two directions the results were identical. In some cases it happened that the rupture took place under a much smaller weight. This, however, could be attributed either to some faults in the structure of the crystal, or to some other disturbances. Such measurements were simply disregarded. Taking as the average value 91*7 lb., or 41*5 kgrs. per square cm., one can calculate the inward pull per row of unit length. Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922. F 66 Breaking Stress of Crystals of Rock- Salt. This is obtained by dividing P by p 1/3 , and becomes P 415000x981 1 . ' P7* = 2'82xl0? =1*4 dynes per cm. If the attraction between tbe charges in a row is not appreciably influenced by the adjacent rows, this value will represent the accurate value of surface tension in the direction coinciding with the vertical axis. If the field is symmetrical in all three directions parallel to the main axis, this figure will characterize the surface tension of rock-salt in all three directions. The symmetrical structure indicated by W. H. and W. L. Bragg is in accord with the experiment in this sense, but on the other hand, the above figure is about 1000 times less than one would expect. One could expect such a small value if the salt consisted of molecules with a very small polarity situated at large distances from one another. But in such a case the force in the direction coinciding with that of the doublet would have to be twice that in the perpendicular direction. This, however, is not the case." In my paper (loc.cit.) it was shown that in a case of small doublets the adjacent rows have practically no influence on the attraction between the doublets. However, in a case of charges situated at regular distances from one another, such seems not to be the case. If the charges influence one another, one can expect the forces to be weaker in the middle of the substance, and much bigger at the surface where the above effect is only one- sided. It is therefore probable that the above figure 1*4 dynes per cm., although quite characteristic for the substance, is not the actual value of the force in the surface layer. The calculation of these effects is not easy, owing to the fact that one has to deal with a very slowly converging series. I satisfied myself that these influences may be appreciable, but I do not see clearly at the present time whether they can account for the weakening of the forces about 1000 times, or even more. 6 Featherstone Buildings, High Holborn, London, W.C. 1. [ 67 ] VII. The Measurement of Chance. By Norman Campbell, Sc.D.* Summary. IT is maintained that the chance of an event happening is always a physical property of a system, measured by a process of derived measurement involving the two fundamental magnitudes — number of events and number of trials. Chances are not measurable by a process of fundamental measurement. But the calculation of chances is analogous to fundamental measurement. It is usually theoretical, and is valuable only in so far as the calculated chances are confirmed by measurement. When a proposition concerns a system characterized by a chance, it may sometimes (but by no means always) be regarded as having a definite probability determined by that chance. The probability of propositions which do not concern systems characterized by chances has nothing: to do Avi tli chance. 1* It is generally recognized that there are two kinds of " probability.'-' There is (1) the probability (of the happening) of events, and (2) the probability (of the truth) of propositions. Etymologically the term belongs more properly to the kind second of probability, and it will be confined to that kind in this paper. For the first kind, the term " chance,''' often used in some connexions as a synonym of probability, is available. Accordingly we shall speak throughout of the chance of an event happening and of the probability of a proposition being true. Various opinions have been entertained concerning the relation between chance and probability and between the methods of measuring them. Some have held that chance, some that probability, is the more fundamental conception, and that the measurement of the less fundamental depends on that of the more fundamental conception. Others have held that only one of the two, or neither, is measurable. The conclusion towards which this paper is directed is that chance, in the sense primarily important to physics, is a phy- sical property measurable by ordinary physical measurement. This view is similar to that held by Venn ; indeed, it is * Communicated by the Author. F2 68 Dr. Norman Campbell on the probably the view that Venn would have held if he had ever considered the nature of: physical measurement. But the further view often attributed to Venn, though it is doubtful whether he actually held it, that probability is always measurable in terms of chance — this view will not be upheld, but, so far as it is discussed at all, will be combated. Many of the ideas and terms used in the discussion are explained more fully in my ' Physics,' to which references are made by the letter P. In fact, this paper may be regarded as a substitute for pp. 168-183 of that book, some of the difficulties of which are avoided by the alternative method of treatment. However, I should like to add that I do not accept any of the criticisms that have been directed against those pages by others. 2. Suppose we are presented with a pair of dice, and asked what is the chance that when one of them is thrown it will turn up six. The answer may be different for the two dice. If one of them is accurately cubical with its centre of mass accural ely at the centre of the cube, while the other has corners and edges variously rounded and is loaded so that the centre of mass is appreciably nearer one face than another, then the answer will be different. On the other hand, if in all respects the dice are the same, then the answer will be the same ; even if they are both inaccurate in form and both loaded — the inaccuracy of form and the loading being the same — the chance that they will turn up six will be the same. This chance is something uniformly associated with and changing with the structure of the die, just as is (say) the electrical resistance. This uniform association of the resistance with the other characteristics of the die is what we assert when we say that; the resis- tance is a physical property of the die, and accordingly the chance of turning up six is a physical property as much as the resistance. Moreover, the chance is measured by essentially the same process as that by which the resistance is measured. Resist- ance is measured (in its original meaning) as a derived magnitude by means of a numerical law (P. Ch. xiii.).. We place two electrodes in contact with opposite faces of the die, and measure the current which flows through it when measured potential differences are maintained between the electrodes. We then plot current against P.D., and find we can draw a straight line through (or more accurately among) the resulting points. The fact that the graph is a Measurement of Chance. 69 straight line passing through the origin shows that a numerical law of a certain form holds, and therefore that the die is characterized by a single definite magnitude, which is what we mean by resistance ; the slope of the line tells us the numerical value of this magnitude. When we proceed to measure the chance of turning up six, we make several groups of trials, measure in each group the number of trials and the number of those in which six turns up. We plot these two fundamental magnitudes against each other, and find that a straight line can be drawn through (or among) the points. The fact that the graph is straight and passes through the origin tells us that the die is characterized by a definite magnitude, which is what we mean by chance ; the slope of the line tells us the numerical value of this magnitude. 3. The resemblance is exact in all essentials. But as the conclusion that chance is an ordinary physical magnitude does not seem to be universally accepted, some objections may be considered. The first may be (though I am not sure that it will be) raised by those who denounce the "frequency theory " of probability. They might say that, though the derived magnitude, estimated in the manner described, is a true or approximate measure of the chance, yet it is not what is meant by the chance — that is something much more abstruse. Such an objection can only be met by stating more clearly what is asserted, and recognizing any difference of opinion that remains as insoluble. What I assert is (1) that all chances determined by experiment are determined by a relation between frequencies, and (2) that chances are important for physics only in so far as they represent relations between frequencies. Few examples can be cited in support of (1), for chance in physics is usually a theo- retical and not an experimental conception ; but it may be suggested that anyone who proposed to attribute to the chance of a given deflexion of an a-ray in passing through a given film any value other than that determined by fre- quency, could convince us of nothing but his ignorance of physics. In support of (2) it may be pointed out that the chance, which is such an important conception in the statistical theories of physics, enters into the laws predicted by those theories only because it represents a relative frequency. 70 Dr. Norman Campbell on the 4. A second objection may be based on the fact that the straight line has to be drawn among and not through the experimental points. It may be readily admitted that this fact shows that the chance cannot be estimated with perfect accuracy. But there is also some uncertainty in determining the resistance ; and since I am concerned only to enforce the analogy between chance and resistance, the admission is innocuous. If it is urged that this uncertainty shows that the derived magnitude cannot be the chance, because chance is something to which a numeral may be attached with mathematical accuracy, then it is replied (as in answering the first objection) thnt such a chance, to which no experi- mental error is attached, is something totally irrelevant to physics. But the objection may be put in a less crude form. It may be urged that, in the matter of experimental error, there is a fundamental difference between resistance and chance. For in the latter, but not in the former, the error is something essential to the magnitude ; we can conceive of a resistance measured without error, but not of a chance measured with- out error ; if all the points lay accurately on the line, then the magnitude measured by its slope would not be a chance. Again, there is a simple relation between the average error about a point on the " chance " line and the co-ordinates of that point ; while in the " resistance " line the relation is much more complex, and depends on the exact method of measuring the current and potential. All this is quite true, and would be important if we were considering the theory of chance or of resistance. There is a great difference in those theories; we suppose that the "real" Ohm's law holds between the real and not the measured magnitudes of the current and potential, while there is no real magnitude involved in the chance relation. But w T e are not considering theory but experiment; I am only asserting that chance is an experi- mentally measured magnitude. The fact that the errors in the two cases are differently explained does not affect the fact that there are errors in both cases, and that the problem of determining the derived magnitude in spite of these errors is precisely the same. 5. As a third objection it might be urged that the two measurements are not really similar, because ihe chance is not really determined by the slope of the line, but by the ratio of the two numbers when they are sufficiently great. Here is a misconception which it is important to correct. If we know that the happening of the events is determined Measurement of Chance. 71 by chance, then it is true that we need only plot one point on the line ; and the distribution of the "errors is such that the relative error of a determination from a single point is less the greater the number of trials involved. We shall group all our observations together, so as to make their total number as great as possible. But similarly, if we know that the material of the die obeys Ohm's law, one observa- tion is sufficient to determine its resistance ; and the accuracy of the determination will be greatest if we choose the measuring current within a certain range. An even closer parallel would be obtained if we took in place of resistance the derived magnitude, uniform velocity. If we knew that the velocity was uniform, we should choose our time and distance as great as possible, and determine the velocity from this single pair of values without troubling to plot smaller values. But in order that determination by a single point should be legitimate, we must know that the events really are determined by chance, and the only test of chance is that, when a series of points are plotted in the manner described, the only regularity discoverable in them is that they lie about the straight line. Their distribution about that line must be random. Thus, to take Poincare's excellent example, if the trials were made by selecting the first figure of the numerals in a table of logarithms in the conventional order, and the events were the occurrence of the figure 1, the plotted points would lie on the whole about a line with a slope of 1/10. But a regularity of distribution about that line would be apparent ; we should have a series of points all lying above the line followed by a set all lying below. If, on the other hand, we took the last figure of the numerals, no such regularity would be apparent ; the distribution of the points about the line would be random ; the events would be dictated by chance. It is of the first importance to insist that in measuring a chance we are picking out the only regularity that we can find in some sequence of phenomena, leaving a residuum which is purely random. Randomness is a primary con- ception, incapable of further definition ; it cannot be explained to anyone who does not possess it. It is based, I believe, on observation of the actions of beings acting consciously under free volition ; and it is subjective in the sense that what is random to one person may not be random to another with fuller knowledge (P. p. 203") . There are certain forms of distributions that are random to everybody ; it is this common randomness, objective in the sense in which all the 72 Dr. Norman Campbell on the subject-matter of science is objective, that is the characteristic of the objective c'hance which is physically measurable. Chance is applicable only to events which contain an element which is wholly and completely random to everybody *. 6. We shall then base our further discussion on the assumption that any physically significant chance (of the happening of an event) is a measurable derived magnitude, a property of the system concerned in that event, determined by a linear numerical law relating the fundamental magni- tudes, number of events and number of trials. It is thereby implied that the " errors " from the law are random, for otherwise the law would not be linear. The definition of chance as the limiting ratio of the fundamental magnitudes as they tend to infinity is identical with that given, if it is known as an experimental fact that the magnitudes of the errors fulfil certain conditions which need not be discussed in detail here ; these conditions are not inconsistent with the randomness of the errors. Chance as a fundamental magnitude. 7. Another important question may be raised, again suggested by the analogy with resistance, Resistance means the derived magnitude defined by Ohm's law. But actually resistance is measured nowadays, not as a derived, but as a fundamental magnitude, in virtue of the Kirchhotf laws for the combination of resistances in series and parallel f. Can chance, though meaning the derived magnitude, be measured independently as fundamental ? In order that a property may be measured as a funda- mental magnitude, it is necessary that satisfactory definitions of equality and of addition should be found (P. Ch. x.). In addition, some numerical value must be assigned arbi- trarily to some one property, which with all others can * In this sense the last figure of the logarithm is not wholly dictated by chance ; for we know that there must be some regularity in the distribution of the points about the straight line, even if we cannot say exactly what it is. In the strictest sense, therefore, there is no such thing as the chance of the last figure being 1. But there are events which are, at present at least, wholly dictated by chance in this sense, e. g. the disintegrations of a radioactive atom. Here I do not think anyone has imagined what kind of regularity there can be, except the falling of the plotted points about the straight line which determines the chance. f Ultimately measured, that is to say, by the makers who calibrate our resistance boxes. In the laboratory we use a method which is essentially that of judgment of equality with a graduated instrument. Measurement of Chance. 73 be compared by means of these definitions. Since we take the meaning of chance to be that of the derived magnitude, the definitions will be satisfactory if they are in accord- ance with the derived process of measurement ; but we shall not succeed in establishing an independent system of fundamental measurement, unless the definitions are such that they can be applied without resort to that process. The arbitrary assignment is usually made by attributing the value 1 to the chance of an event which always happens as the result of a trial. The only question that can arise here, namely whether all other chances can be connected with this chance by addition and equality, will be considered presently. The definition of addition presents no difficult}'. The chance of A happening is the sum of the chances of .r, y, z, ... happening if, x, y, z, ... being mutually exclusive alternatives, A is the event which consists in the happening of either x or y, or z, .... This proposition is introduced in all discussions of chance, but it is often introduced as a deduction and not as a definition. The inconsistencies which result from such a procedure are discussed in P. pp. 174, 184, 185. As a definition, it is satisfactory in our sense, for measurements by the derived method would show that, in such conditions, the chance of A is the sum of the chances of x, y, z, ..., and yet it does not presuppose such derived measurements. If the points in the derived measurement lay on the straight line, this result would be a direct consequence of the definition of the derived magnitude ; but since they do not, it can be deduced from that definition only if some assumption about the distribution of the errors is made. The assumption that the errors are random would probably suffice if randomness could be strictly defined ; since it cannot, the agreement of the proposed definition of addition with the results of the derived process of measurement must be regarded as an experimental fact. The definition is thus precisely analogous to that used in the fundamental measurement of resistance, namely that resistances are added when the bodies are placed in series. The definition of equality is much more difficult; in fact, it is the stumbling block of many expositions of the measure- ment of chance. For resistances we can say that bodies are equal if, when one is substituted for another in any circuit, the current and potentials in that circuit are unchanged ; that definition does not involve a knowledge of Ohm's law and of the derived measurement. The only attempt at an analogous definition for chance, of which I am aware, is that based on the principle of sufficient reason ; chances 74 Dr. Norman Campbell on the are said to be equal when there is no reason to believe that one rather than the other will happen as the result of any trial. But what reason could there be for such a belief based on experiment ? No a priori principle can determine the property of a system, which is an experi- mental fact ; we cannot tell whether a die is fair or loaded without examining it through our senses. The only experi- mental reason I can conceive for believing that one event is more likely to happen than another is that it has happened more frequently in the past. But if an attempt is made to define " more frequently " precisely, the judg- ment of equality is inevitably made to depend on the derived measurement, and the fundamental process ceases to be independent of it. This dependence is often con- cealed by the use of question-begging words. Thus, the principle of sufficient reason may be reasonably held to decide that, in a perfectly shuffled pack of cards, the chance that the card next after a heart is another heart is equal to the chance that it is a club. But if inquiry is made what is meant by a perfectly shuffled pack and how we are to know whether a pack is or is not perfectly shuffled, I can seen no answer except that it is one in which a club occurs after a heart as often as another heart. But, of course, to define perfect shuffling in that way is to admit that the criterion of equality is based upon the derived measurement of " frequency " *. I can find no proposed definition of the equality of chances that is both applicable to experimental facts and independent of frequency ; and I conclude, therefore, that there is not for chance, as there is for resistance, a fundamental process of measurement independent of the derived. 8. But there is a further difference to be considered. Even if equality of chance could be defined independently, there would still be many chances (and those some of the most important) which could not be connected with the unit by the relations of equality and addition: Any resistance is equal to the sum of some set of resistances such that the sum of another set of them is equal to the unit or to the sum of some set of units. The analogous proposition about chances * It is not always realized by those who calculate card chances in great detail that in actual play, even among experienced players, the shuffling is so imperfect as to distort very seriously the chances of such events as the holding of a very long suit. Measurement of Chance. 75 is not true, if the chances are always experimentally deter- mined. Consider, for example, the disintegration of a radioactive atom within a stated period. There are only two alternatives : the atom does disintegrate, or it does not. The sum of the chances of these two events is 1, but the chances of the two are not in general equal. And neither of them can be shown experimentally to be the sum of the equal chances of other events such that the sum of some other set of those chances is equal to the unit. The definition of unit chance together with definitions of equality and addition would never permit us to determine such chances ; they can only be determined by derived measurement. 9. Chance is therefore not capable of fundamental measurement. Nevertheless the principles of fundamental measurement are important in connexion with chance, because they are involved in the calculation of chances. When we calculate a chance we always assume that it is measurable by the fundamental process. Thus if we calculate the chance of drawing a heart from a pack, we argue thus : — The chance of drawing any one of the 52 cards is equal to that of drawing any other. The chance of drawing one of the 52 cards is, by the definition of addition, the sum of the chances of drawing the individual cards, and, by the definition of unit chance, it is 1. Consequently the chance of drawing any one card is 1/52. But the chance of drawing a heart is the sum of the chances of drawing 13 individual cards ; it is therefore the sum of 13 chances each equal to 13/52, i. e. 1/4. The calculation is perfectly legitimate, so long as we know (1) of how many individual events the event under consideration (and any other event introduced into the argument) is the sum, and (2) that the chances of these individual events are equal. (1) does not depend on the derived system of measurement, but it does involve a very complete knowledge of the event under consideration; (2), if it is an experimental proposition at all, must depend upon derived measurement. The calculation is often made when (2) is not experimental, and when there is no direct know- ledge of (1) ; it is then purely theoretical, and the only legitimate use that can be made of it is to confirm or reject the theory by means of a comparison of the calculated chance with that determined experimentally by the derived measure- ment. The fact remains that true chance, the property of the system, is always and inevitably measured by the derived process and not by the fundamental. 76 Dr. Norman Campbell on the Chance and Probability. 10. It remains to consider very briefly what connexion, if any, there is between the chance of events and the proba- bilities o£ propositions. Probability is usually admitted to be an indefinable conception, applicable to propositions concerning which there is no complete certainty, and roughly describable as the degree of their certainty. It appears to me one of those conceptions which are the more elusive the more they are studied ; I am quite certain that I do not understand what some other writers mean by the term, and am not at all certain that I can attach a perfectly definite meaning to it myself. The observations that I can offer are therefore necessarily tentative. But it is clear, at any rate, that probability is not a property of a system and is not physically measurable ; any propositions connecting it with chance must depend ultimately on fundamental judgments which can be offered for acceptance, but cannot be the subject of scientific proof. There are two kinds of propositions the probability of which may plausibly be connected with chance ; and they naturally can apply only to systems that are characterized by chances. Of the first kind the following is typical : — This die will turn up six the next time it is thrown (or on some other single and definite occasion). Here (ef. P. pp. 192-200) it seems that, if the proposition is really applied to a single occasion only, the probability of the proposition must be that characteristic of absolute ignorance ; for the assumption that anything whatever is known of the result of a single trial is inconsistent with the experimental fact that the result of any one trial is random. The only exception occurs when the event is one of which the chance is so small (or so great) that the happening of it (or failure of it) would force us to revise our estimate of the chance or to deny that there was a chance at all. Of such coincidences, in systems of which the chance has been well ascertained, the assertion that they will not (or will) occur may be made with the certainty that is characteristic of any scientific statement. There is no probability. On the other hand, it is very difficult to be sure that only a single trial is contemplated. For when such statements are important, there is always a clear possibility of a consider- able number of repetitions of the trial. If this number is so great as to permit a dermination, by derived measurement, of the chance of the event within some limits relevant to the Measurement of Chance. 11 problem, then it will be found by examination of the use of such propositions that their importance depends simply on the value of that chauce. If that chance is greater than a certain value, the proposition will be true for the purposes concerned ; if it is less, it will be false. I cannot myself ever find in such propositions any meaning which is not contained in the proposition : — The chance that the die will turn up six is greater or less than some other chance. Accordingly again, there seems no room for a probability which is distinct from chance. 11. Of the second kind of proposition an answer to the following question may be taken as typical : — I have two dice, of which the chances of turning up six are unequal. I throw one, but I do not know which. It turns up six. Which oE the two dice have I thrown? Here again (P. pp. 185-192), if the question is asked of a single throw, it seems to me that the only possible answer is simply, I do not know ; except, as before, if the throw would be a ei coincidence " with one die and not with the other. For, once more, if the events concerned are really characterized by chances, it is inconsistent with the statement that they are so characterized to assert that, at a single trial, the result, if compatible with either of the two " causes/' may not happen as the result of either of them. If, on the other hand, the throws are repeated (while it is certain that the same die is always used), and if they deter- mine the chance of one die rather than that of the other, it is clearly certain that this die, and not the other, is being- used ; a die can be identified by its chance as certainly as by its resistance or any other physical property. But intermediate between these extremes, there certainly seem to be cases in which, though the evidence is not sufficient to enable us to assert definitely which die is being thrown, we begin to suspect that it is one and not the other. The possibility of such a state of mind arises from the fact that there is necessarily a finite period during which the evidence is accumulating ; it does not arise when, as in the usual determination of resistance, the evidence is obtained all at the same time. And our suspicion will increase generally with the " probability " as estimated by the well- known Bayes's formula for the probability of causes. In this case it appears to me that there is such a thing as probability, determined by but distinguishable from chance, and applying to a proposition, and not to an event. But I can find no reason to believe that this probability is 78 The Measurement of Chance. numerically measurable in accordance with Bayes' or any other formula. But in most cases where an attempt is made to apply a probability o£ causes, the condition is not fulfilled that it is known that the same die is always used. If that condition is not fulfilled, the probability, according to orthodox theory, depends on certain a priori probabilities which are not chances. The problem then ceases to be one of the connexion between chance and probability, and thus falls without the strict limits of our discussion. 12. But it is necessary to transgress those limits for one purpose.' It has been often urged by philosophers that probability is characteristically applicable to scientific propositions, which are to be regarded, not as certain, but ouly as more or less probable. If this be so, the con- ception of chance, being a scientific conception deriving its meaning from scientific propositions, must be subsequent to the conception of probability, and the order of our discussion should have been reversed. Of course I do not accept the philosophical view, and perhaps it will be well to explain very briefly why I reject it. Doubtless there is a sense in which scientific propositions are not certain ; but in that sense no proposition is certain, so long as its contrary is comprehensible. For if I can understand what is meant by a proposition, I can conceive myself believing it. I am not perfectly certain either that Ohm's law is true, or that (x + a) 2 — a? -\- Zax -\- a 2 : I can conceive myself disbelieving either. If I were forced to say which I. believe more certainly, I should choose Ohm's law ; for I could give a much better account of the evidence on which I believe it. A mathematician, of course, would make the opposite choice. But it appears to me useless to com- pare the " certainties" of two propositions when they are of so different a nature that the source of the uncertainty is perfectly different. If a proposition is as certain as any proposition of that nature can be, and if nothing whatever could make it more certain, then it seems to me misleading to distinguish its probability from certainty. Now, fully established scientific propositions are ecrtainin this sense. They are uncertain only in so far as they predict If in asserting Ohm's law, I mean (and I think this is my chief meaning) that it appears to me a perfectly complete and satisfying interpretation of all past experience and that other people appear to share my opinion, then Ohm's law is Ignition of Gases by Sudden Compression. 79 perfectly certain, or at least as certain as any mathematical or logical proposition. On the other hand, it' it is meant that the law will be to me and to others an equally satisfying interpretation of all future experience, then I am not absolutely certain ; I am only as certain as I can be about anything in the future. And it must be noticed that nothing- can make me more certain. If I were predicting something about a single future occasion, I might in the .course of time become more certain; for that future occasion might some day become past. But if, as in the case of a scientific law, I am predicting something about all future experience, then, since the future is indefinite, no amount of additional experience, converting finite portions of the future into the past, can make me more certain; for there will always remain as much future as before. Such uncertainty as there is in the proposition is inherent in its nature ; if it were absolutely certain, it would not be the same proposition. VIII. Experiments on the Ignition of Gases by Sudden Compression. By H. T. Tizaed and D. R. Pye *. [Plate I.] I. TN a previous paper f, it was shown that when a mixture JL of a combustible gas or vapour with air was suddenly compressed, explosion miglit take place after an interval the duration of which depended on the temperature reached by the compression. It is known that below a certain tempera- ture, called the ignition temperature, no explosion, and no very appreciable reaction, takes place under these conditions ; and the experiments referred to showed that just above the ignition temperature, the delay before explosion occurs may be of the order of one second in certain cases, while — in the case of hydrocarbons and air — the delay at a temperature some 50° above the ignition temperature was very small. It was pointed out that the observed ignition temperature must not only depend on the properties of the combustible substances, but also on the conditions of experiment, and particularly on the rate of loss of heat from the gas at the * Communicated bv the Authors. t H. T. Tizard, "The Causes of Detonation in Internal Combustion gines." Proceedings of the N.E. Coast Institution of Engineers and Shipbuilder, May 1921. 80 Messrs. H. T. Tizard and D. R. Pye on the ignition temperature. The fact that this has not been fully taken into account previously seems to account in some measure for the differences in the results obtained by other workers. It was further shown that the period of slow combustion before explosion took place also depends on the properties of the combustible substance, and a theory was briefly developed connecting the "delays" observed at different temperatures with the effect of a rise in temperature on the rate of combustion, i. e. with the so-called tempera- ture coefficient of the reaction. The object of the experi- ments described in this paper was to test these theories quantitatively, and to attempt to deduce from the results the temperature coefficients in certain typical cases. The mea- surement of the temperature coefficients of simple gaseous reactions is of considerable importance in connexion with the theory of chemical reactions,, for one of the great difficulties in the development of theory hitherto has been the fact that most gaseous' reactions have to be investigated under conditions which are complicated by the disturbing influence of solid catalysts or of the walls of the containing vessel. Gaseous reactions which occur on sudden com- pression are free from this complication, for the walls of the containing vessel are much lower in temperature than the gas; by quantitative measurements of the rate of loss of heat near the ignition temperature, and of the delay before explosion occurs, it therefore seems possible to gain some real insight into the mechanism of homogeneous gas reactions. Experiments of this nature also have a considerable practical interest for the development of internal combustion engines, for, according to our views, the tendency of a fuel to detonate at high temperatures depends not only on its ignition temperature but also on the temperature coefficient of its reaction with oxygen. Previous experiments on the ignition of gases by sudden compression have been made by Talk, at Nernst's suggestion (J. Amer. Chem. Soc. xviii. p. 1517 (1906), xxix. p. 1536 (1907)), and by Dixon and his co-workers (see Dixon, Brad- shaw & Campbell, Journ. Chem. Soc. 1914, p. 2027 ; and Dixon & Crofts, p. 2036). Nernst first put forward the view that at the ignition temperature the evolution of heat due to the reaction was just greater than that lost to the sur- roundings; but this suggestion has not hitherto been carried further, since no previous workers have attempted to measure the rate of loss of heat near the ignition tempera- ture. Further, previous work has been mainly confined to the measurement of ignition temperatures of mixtures of Ignition of Gases by Sudden Compression. 81 hydrogen, oxygen, and an indifferent gas. In such experi- ments the interval which occurs at the lowest ignition temperature between the end of the compression and the occurrence of ignition is very small ; under these conditions an apparatus of the kind used by Nernst and Dixon gives fairly satisfactory results. It is not well suited, however, for experiments with other gases, such as the hydrocarbons, when there may be an appreciable delay before ignition occurs. In such cases it is of great importance to ensure that the cylinder in which the compression is effected is as gas-tight as possible, and that the piston is held rigidly in position at the end of the compression stroke. II. The apparatus used for our experiments was originally designed and built by Messrs. Ricardo & Co. with a view to determining the temperatures of spontaneous ignition of various fuels used in internal combustion engines under conditions which correspond closely with those obtaining in an engine cylinder. ria 1 Fio-. 1 shows diagrammatically the arrangement of the mechanism. A very heavy flywheel A rotates quite freely on the shaft B, and is kept spinning by an electric motor at about 360 R.P.M. The shaft B carries between bearings the crank D, and outside one bearing, the internal expanding clutch C, which can engage with the flywheel rim. The piston E moves vertically in the jacketed cylinder F, which has an internal diameter of 4J inches and can be raised or lowered bodily in the heavy cast-iron casing of the apparatus when the compression ratio is to be altered. The length of stroke of the piston is 8 inches, and its motion is controlled by the two hinged rods Gr and H of which the latter is carried on a fixed bearing at K. The hingo L is Phil. Mag. S. 6. Vol. 44. No. 259. July 1922. G 82 Messrs. H. T. Tizard and D. R. Pye on the linked up with the crank pin by the compound connecting rod N. That part of the connecting rod attached to the crank pin is tubular and contains the sliding rod M attached to the hinge L. A clip carried on the sleeve can engage with a collar on the inner rod and hold the latter rigid in the tube. With the connecting rod locked as one link, the crank is rotated by hand for setting the piston in its lowest position. When a compression has to be made, the clutch is suddenly expanded by a hand lever while the flywheel is running at high speed, clutch and crank are carried round with the flywheel, and the toggle joint ELK is straightened until the hinge L lies on the vertical line between the piston centre and hinge K. At the moment when L is vertically over K it comes up against a leather pad, and a clip comes into action which holds it in this position. At the same moment, too, the clip releases the two parts of the com- pound connecting rod, so that while the two rods G and II are held in the vertical position to take the large downward thrust of the piston when explosion of the compressed mixture occurs, the crank, clutch, and flywheel are free to go on rotating, and the shock due to destruction of the momentum of the moving parts is reduced to a minimum. The initial temperature of the gases in the cylinder can be varied by means of a water jacket round the cylinder, and the variation of pressure during and after compression is recorded by means of an optical pressure indicator of the Hopkinson type. For the purposes of the present experiments it was necessary : — (1) To make the compression space above the piston absolutely air-tight so as to eliminate all pressure drop due to leakage. (2) To arrange an accurate timing gear upon the indicator so that pressure-time records correct to about one per cent, could be obtained. (3) To insert some kind of fan in the compression space so that the effect of varying turbulence in the compressed gas could be determined. (1) Air-tightness of the Compression Space. There were three points at which appreciable leakage was liable to take place : — (a) Round the sides of the main piston. (b) Past the indicator piston. (c) Round the fan spindle. Ignition of Gases by Sudden Compression. 83 Of these, the first two were completely eliminated and the third reduced to something quite negligible. Figures of the actual leakage are given below. The method of eliminating piston leak will be understood from fig. 2. The piston is made in two parts. Below the F/a 2 cast-iron top, which carries a single piston ring of the usual type, there is an aluminium body which for some distance is of smaller diameter than the cylinder. Round this waist are two cup-leathers, C and D, each with the periphery turned upwards, which are separated by a cast-iron ring of square section (E, fig. 2). The whole space above and between the cup-leathers was filled with castor oil so that no air could leak down past the piston head until it had first made a space for itself by forcing castor oil down past the two cup-leathers in series. ¥3r g .JP2 i A na 3 To eliminate le k past the indicator piston, the plan was hit upon of turning the whole indicator upside down and then pouring in a little castor oil into the space above the piston, as illustrated in fig. 3. Here A represents the piston of the inverted indicator with the sp;ice above it filled with G2 84 Messrs. H. T. Tizard and D. R. Pye on the castor oil. The duct BC in the bracket which carries the indicator is based on the slope to prevent the possibility of any oil getting back along it and into the combustion space at C, This arrangement was very satisfactory, for, besides forming a perfect seal, it served to keep the piston well lubricated ; it was found, moreover, that only a very small quantity of oil was forced down past the piston, even after continued exposure to high pressure. The arrangement of fan and fan-spindle is shown in fig. 4. Here the plug A, shown in section, was made to screw vertically* downwards in the centre of the cylinder LEA THEH head, its lower end finishing flush with the top of the com- pression space. At the highest compression ratios this space is \ inch deep, so that the piston head under these conditions is close up to the under side of the fan. The fan was driven at varying speed by an electric motor through the usual form of flexible drive used for speedometers. To prevent leakage up round the fan-spindle, this was made long and thin, and was also provided with a cup-leather of which the tightness could be adjusted by the screw B in fig. 4. By using a motor with ample reserve of power for driving the fan, it was possible to keep the cup-leather so tight as to reduce the leakage to a negligible amount. Ignition of Gases by Sudden Compression. 85 Experiments on Air Leakage from the Cylinder. Observations were made of the rate of fall of pressure beginning 10 seconds after compression had occurred. By this time all pressure-fall due to heat-loss has ceased, and the observed fall represents leakage only. Experiments were made — (a) With fan stationary. Compression ratio 9 : 1. Contents of cylinder were the products of com- bustion of a benzene and air mixture. rp. Deflexion on Indicator e * Screen. 10 sees, after compression and combustion 063 inches. 20 „ „ „ 0-61 „ 30 „ „ „ 0-57 „ 45 „ „ „ 0-56 „ 60 „ „ „ 0-53 „ The indicator calibration was 1 inch deflexion = 188 lb. per sq. in., so that the above gives a rate of fall of 0-1 X 188 >aftll — = -5b lb. per sq. in. per sec. (b) With fan rotating at full speed, 2000 r.p.m. Com- pression ratio 10 : 1. Contents of cylinder pure hydrogen. m . Deflexion on Indicator Time. c Screen. 10 sees, after compression 066 inches. 20 „ „ 053 „ 30 „ „ 0-40 „ This, owing to the low viscosity of hydrogen and the high compression ratio, was a very severe test. The rate of fall was '26 x 188 AK „ . xt ■ .u- — r = 2*45 lb. per sq. in. per sec. JNow in this expe- riment the rate of fall of pressure due to cooling was as high as 600 lb. per sq. inch per second immediately after com- pression, so that the effect of. gas leakage on the apparent rate of cooling was clearly negligible. The Indicator. It will be convenient at this point, before going on to describe the timing apparatus, to give some further data as to the indicator used. As stated above, this was of the 86 Messrs. H. T. lizard and D. R. Pye on the standard Hopkinson optical type *, and need not be described in detail. The piston used throughout the experiments was one of 0*125 sq. in. area. Pressure on the piston deflects a spring attached to it, and thus tilts a small mirror which reflects a point of light from a fixed lamp. The magnitude of the pressure is thus arranged to correspond to the vertical (downward) deflexion of the image of the light on a 4£ x 3^ photographic plate carried in a camera fixed to the indicator. The calibration of the pressure scale was made in two ways. Firstly, by subjecting the piston to known oil pressures and measuring deflexions on the camera screen, and, secondly, as a check on the piston area, by direct dead- weight loading of the spring and measurement of the deflexion. The oil pressures in the first calibration were produced by a carefully gauged vertical plunger loaded with weights. The calibration by this method was carried out at the beginning of the experiments, and it was found that the relation between pressure and deflexion was given very closely by a straight line at a slope corresponding to a pressure rise of 188 lb. per sq. in. per inch deflexion. At the conclusion of the experiments this calibration was checked by the dead-weight calibration as follows :; — Keading on Camera Screen. Load on Spring. Loading. Unloading. 0-39 in. 040 in. Weight carrier = 325 lb 0-54 „ 0-54,, Carrier + 10 = 13-25 „ 0*975 „ 0"99 „ „ + 20 =23-25 ,', 1-40 „ 143 „ „ + 30 =33-25 „ 1-81 „ 1-81 „ This also shows a satisfactorily straight-line relationship which checks remarkably well with the previous calibration, for '^^'9 Dead weight per inch deflexion = , ~ — 234 lb. corresponding to a pressure on a piston of area 0*125 sq. in." of 8 x 23-4 = 187-2 lb. The previous figure of 188 lb. per sq. in. per inch de- flexion has been used throughout the calculations. (2) The Timing Gear. The pressure in the cylinder being given by the vertical motion of a point of light on a photographic plate, it was necessary for the accurate measurement of the lengths ot the delay period to give the point of light at the same time * Hopkinson, Proc. Inst. Mech. Eng. Oct. 1907, p. 863. Ignition of Gases by Sudden Compression. 87 a uniform and exactly known motion across the plate horizontally. To obtain this uniform horizontal velocity, the arrangement in fig. 5 was adopted. A is a vertical cylinder closed at the bottom and carrying at the top a guide for the piston rod B. At the lower end of the rod B is a light loosely fitting piston in which are drilled one or more small holes. The rod B has a collar C above the guide, and can be loaded with weights, up to 50 lb. if need be. The cylinder is filled with paraffin which, when the piston and weights have been raised and then released by a trigger arrangement, is forced at high velocity through the holes in the piston from the under to the upper side. This arrangement gives a velocity of fall of the piston which is uniform to within 1 or 2 per cent., and it was found, moreover, that this velocity varied directly as the square root of the weight carried : a result which shows that the viscosity of the oil is a negligible factor in deter- mining the rate, and that the latter will therefore not be affected by any small changes of temperature which might occur from day to day in the oil. This point was checked experimentally, and it was found that the maximum change in the rate of fall produced by heating the oil from 19° C. to 65° 0. was 1|- per cent. As the oil temperature during the experiments never varied by more than 2° or 3°, the effects of temperature changes were quite negligible. Piston and weights are suspended by a steel wire from an arrangement of pulleys (F) carried on spindles attached to 88 Messrs. H. T. Tizard and D. R. Pye on the the cylinder head. D and E are balance weights to keep the wires taut, and F is a compound pulley which reduces the horizontal motion o£ the wire Gr to about one-half the vertical movement of the piston. At H the wire Gr is divided, and each half wraps round the periphery of a sector attached to the arm K of the indicator. The barrel of the indicator is thus uniformly rotated through a sector about a vertical axis, and the speed of the point of light horizontally across the photographic plate in the indicator camera can be varied between wide limits by alteration of the weights and the number of holes in the loaded piston through which the paraffin is forced. Actually it was found in these experiments that only two speeds were required, of 5*77 and 4* 30 cm. per second. These were obtained when the piston carried weights of 40 and 24 lb. respectively, and was pierced by a single hole £ inch diameter. Both the speed and its uniformity were measured by in- terrupting the light at known intervals while it traced a straight horizontal line across, the plate. To obtain the uniform interruptions, 60 equally spaced contacts were arranged round the periphery of a disk about 6 inches in diameter. An electric motor was used to rotate the disk through a friction clutch, the speed being kept steady by the operation of a governor, which caused the disengagement of the clutch when the speed tended to increase. The arrangement was very similar to that of a gramophone motor, except that the speed of the latter is kept steady by means of a little brake which is operated by a governor, whereas in our experiments the governor operated the clutch. This apparatus, which was made at the Royal Aircraft Establishment, Farnborough, was lent to us for these experi- ments by kind permission of the Director of Research, Air Ministry. To calibrate the falling weight apparatus, the speed of revolution of the disk was first adjusted roughly to one com- plete revolution in 2 seconds. It was then timed repeatedly over twenty complete revolutions by stop-watch ; the mean time for this number of revolutions was found to be 36*5 seconds, the variations in successive timings not exceeding •4 second. Hence, since sixty contacts were made each revolution, the time between the beginnings of successive 36*5 contacts was -=7* ztt; = 0*0304 second. The 6-volt lamp of 60 x 20 r the indicator was then connected up, through the contacts on the disk, to a 12-volt battery ; it was thus greatly " overrun " whenever contact was made. The effect of Ignition of Gases by Sudden Compression. 89 overrunning the -lamp in this way was to make it flicker brightly, so that when the indicator mirror was rotated by the dropping weight, the reflected light made a horizontal line of dashes across the photographic plate, the interval between the middle of two consecutive dashes corresponding to the interval between two contacts on the uniformly rotating disk. The magnitude and uniformity of the velocity of rotation of the indicator mirror by the falling weight could thus be measured. Records were taken with falling- weights of 8, 24, and 40 lb. To show the uniformity of motion during descent, the results of measuring two plates taken with the 24-lb. load are given in full. The apparent variations in any single experiment are due rather to the difficulty of estimating the centre of a flicker to 0*1 mm. than to any real variation in the rate of fall of the loaded piston. Plate B. 0-65 cm. 1-30 1-95 2-6 3-28 395 4-58 5-22 585 651 7-12 7-78 8--10 9-06 9-70 Mean 1 flicker in 0132 cm. Mean 1 in 0*129 cm/ 1 flicker = •0304 second. . * . 1 second = 4*34 cm. on plate. 1 second — 4*26 cm. Mean 4'30 cm. = 1 second. Similar records with loads of 8 and 40 lb. gave time scales of 2*22 cm. = 1 second and 5*77 cm. = 1 second. The means for the two plates with 40-lb. load were 5*75 and 5*79 cm. per second, so that it may safely be assumed that the time scale was known correctly to within 2 per cent., and probably less. This is a degree of accuracy as great as that with which,, as a rule, it is possible to measure the records. The vertical movement of the weights was about 7 inches, Plate A. 5 flickers in 0*70 10 1-34 15 2-0 20 266 25 330 30 3'98 35 463 40 529 45 5-93 50 659 55 7-22 60 7-90 65 8-53 70 9-21 75 9-89 90 Messrs. H. T. Tizard and D, R. Pye on the which produced a horizontal movement of the point of light more than twice as great as was necessary to traverse the photographic plate, so that ample time was provided for the speed to have become uniform before a pressure record was taken. The fact that the indicator and all gear except the actual falling weight was carried on the cylinder head, made it possible to raise or lower this when changing the com- pression ratio without affecting the timing gear. The only difference was an alteration of about an inch in the distance fallen by the weights between the position for maximum and minimum compressions. Figs. 6 and 7 are prints taken from two typical records. Fiff. 6. I J OS 7iW Secon3s o-ys- In the first there is no explosion, and the record shows simply the rise of pressure due to compression, and subsequent fall as the compressed mixture cools. Fig. 7 shows the first rise of pressure due to compression, the " delay " period of nearly constant pressure followed by the practically instan- taneous rise on explosion, and finally the rapid cooling of the intensely hot products of combustion. It may be mentioned that the spring of the indicator was protected from the force of the explosion pressure by a stop which prevented the deflexion from ever being greater than just to the edge of the photographic plate. J II. If a gas at a known temperature and pressure is suddenly compressed in a gas-tight cylinder, we can calculate from a measurement of the maximum pressure reached the average temperature of the gas at the moment of maximum Ignition of Gases hy Sudden Compression. 91 compression. If the volume then remains constant, measure- ment of the rate of fall of pressure with time gives the rate of loss of heat, if the specific heat of the gas is known. One would expect this rate to be closely proportional at any moment to the difference in the average temperature of the Ficr. 7. £ 400 £ 300 .C; 200 (B.I.) /ffnit/on Max/mum Oe/ay=t fa/7 compre ?s/o/? J? ■ /W Te/r?p.3/0°C. seconds, speeo'. 0-5 Time in Secor/o's. gas and that of the cylinder walls. If the temperature of the gas is T° absolute, and that of the walls is 0, then §=<*-">> (i) where " a " is a constant which we call the cooling factor, and which depends on the nature of the gas, and its degree of turbulence in the cylinder. The results given later will be seen to justify this equation. Integration of (1) gives i T o -0 (2) where T is the average temperature when t seconds has elapsed from the moment the average temperature was T . 92 Messrs. H. T. Tizard and D. P. Pye on the Since the total volume remains constant, this equation can be written kg." f P-P, at (?) f if the simple gas laws hold. P max . is here the maximum compression pressure, P the observed pressure t seconds after the maximum compression pressure is reached, and P/- the final pressure of the gas when its temperature is the same as that of the walls. P/is therefore equal to rxPi, where r is the compression ratio and P x the initial pressure (in these experiments one atmosphere) before compression. Fig. 6 is a typical cooling curve obtained when air at an initial pressure of 14*73 lb. sq. in. and temperature of 23° C. was compressed in the ratio of 7*02 to 1. The time of compression was 0*08 second, and the values of the cooling factor a, obtained from observations of the fall in pressure, are given in the following table : — Table I. Time after Observed P -P* , -""max. *-f „ lo s p p* ~ s - 2-3S max. pressure pressure = P P-fy a =~T' (seconds). lb./sq. in. r r f 199-7 96-3 •173 185-7 82-3 •068 •90 •347 174-6 71-2 •131 •87 •52 165 61 6 •194 •86 •695 158-3 54-9 •244 •81 •87 152-4 49 •294 •78 Me? m=-85 6 (initial temp.) = Pi = 14-73 lb./ 23° C. Pf=PiX 702 =103-4. sq. in. Max. temp, (calc.) = 299° C. It was found as a rule that the calculated cooling factor tends to diminish as " t' 3 increases. This may be due in part to errors of observation, for the errors are big when the time interval after the attainment of maximum pressure is small : but experimental error will not wholly account for it, and it may be explained on the reasonable assumption that there is a fair degree of turbulence in the gas just after a sudden compression, which dies down after a short time. Potation of the fan at a high speed increases the rate of cooling con- siderably ; experiments with air gave a cooling factor about Ignition of Gases by Sudden Compression. 93 three times as great when the fan was running as when the gas was " stagnant." This will be referred to later when discussing the results of the experiments of ignition. Raising the compression ratio also increases the cooling factor for air. This is also to be expected since the distance between the top of the piston and the head of the cylinder is lessened. The cooling factor for " stagnant " air in the apparatus was found to increase from about 0*76 at a compression ratio of 6 to 1 to about 1*0 at a compression ratio of 10 to 1, the distance between the piston and cylinder head being approxi- mately 4 cm. (1*6 in.) at the lower and 2*3 cm. ('9 in.) at the higher compression ratio. The experimental error in the cooling factors, obtained by experiments similar to that quoted above, is probably about 5 per cent. It should be noted that the rate of loss of heat under the above conditions is considerable. For instance, in the experiment quoted the maximum difference in temperature between the gas and the walls after compression is 276° ; and since the specific heat c v is about 0*18, the air is losing heat at the maximum temperature at the rate of 0'18 x '85 x 276 = 42 calories per second per gram. It follows that if an explosive mixture of gases is suddenly compressed to its ignition temperature, in such an apparatus as that described above, the initial rate of the chemical reaction at the lowest temperature at which ignition is ob- served must be considerable, for the evolution of heat due to the reaction must equal approximately the rate at which heat is lost to the walls. For example, the total heat of com- bustion of a mixture of a paraffin hydrocarbon and air, in the correct proportions to burn to C0 2 and water, is approximately 700 calories per gram of mixture. If it ignites when suddenly compressed to such a temperature that the rate of loss of heat is 35 calories per gram, the reaction, if it continued uniformly at the initial rate, would be complete in 20 seconds. This illustration may serve to show the general nature of the reactions that occur on sudden compression ; what occurs in practice is that the gas, or part of the gas, reacts so that the evolution of heat takes place at a somewhat higher rate than the loss of heat by conduction etc. ; hence the temperature of the reacting gases must automatically increase, and with it the rate of reaction, until the gas " explodes." The interval between the end of compression and the explosion must clearly depend mainly on three factors : (a) the compression tempera- ture, (b) the temperature coefficient of the reaction, (c) the rate of loss of heat to the walls. 94 Messrs. H. T. Tizard and D. B. Pye on the IV. It is now generally recognized that the rate of an ordinary chemical reaction varies with the temperature in a way which may be empirically expressed by an equation of the form _B k=Ae T , (4) where k is the velocity constant, and A and B are constants, B being the temperature coefficient. In the case of reactions evolving heat, we can write this : _B Q=A« T , (5) where Q is the initial rate of evolution of heat when a definite mixture is suddenly compressed to the temperature T. the lowest ignition temperature T we have, B At Qo = A? (6) where Q can be measured from observation of the rate of cooling of the gases at a temperature slightly below the ignition point : i. e., Qo=«*.(T»-0), (7) where a is the cooling factor, c v the specific heat of the mixture, and 6 the temperature of the walls. Now suppose that the gases are compressed initially to a higher temperature T : the initial rate of loss of heat will then be higher than Q , namely but since the effect of temperature on the rate of evolution of heat due to the reaction is so much greater than that on the rate of loss of heat, it is sufficiently accurate at present to assume for our purposes a constant initial rate of loss of heat = Q . The initial rate of reaction at the higher temperature will be given by Q T = A<?~ 7E \ From (5) and (6) we have ^'ii 1 -^- • • • - (8) where b = 0*4343 B. Now under these circumstances the net rate of gain of heat is (Qt — Qo) calories per second. Ignition of Gases by Sudden Compression. 95 The initial rise of temperature is therefore given by dT „ ^ r, /Qg §-Q.-Qo=Qo(J-i) ... (9) /n 1 dT=^dt, (Ht _ -, \ c v or But since Q = ue v (T — 0), we have 7Q^-<|) = a (VV- (10) V. Can this equation be used to determine the delay, or period of slow combustion, that occurs before the temperature suddenly rises very rapidly, i. e. before explosion takes place ? Strictly speaking, this could only be done if Q, the rate of evolution of heat during the initial reaction, depended only on temperature and not on the concentration of the reacting substances. According to all ordinary theories of reaction, this would not be true ; the rate of reaction should depend in some way on the concentration of the reactants. If we consider such a reaction as the combustion of heptane C 7 H 16 + 110 2 = 7C0 2 + 8H 2 0, we can hardly suppose it is necessary, before the initial re- action — whatever is its nature — can occur, for 1 molecule of heptane and 11 molecules^ of oxygen to collide ; but it is reasonable to assume that the rate of reaction must at least be proportional either to the concentration of the heptane or to that of the oxygen, or to the product of the two. It is necessaiw, however, to point out that there is little, if any, satisfactory evidence that homogeneous reactions in gases obey what is ordinarily understood by the law of mass action. In fact, evidence from the ignition of gaseous mixtures by sudden compression points rather to the reverse. Dixon and Crofts' experiments * on the ignition of mixtures of hydrogen and oxygen are difficult to explain by any reasonable assumption as to the mechanism of the reaction based on the * Dixon & Crofts, Trans. Chem. Soc. 1914, p. 2036. 96 Messrs. H. T. Tizard and D. R. Pye on the law of mass action, even after taking into account the possible effects of a different rate of loss of heat in the mixtures with which they experimented ; the authors state, in fact, that they "can offer no satisfactory explanation of the phenomena observed." Recent experiments by one of us * have shown that the ignition temperatures of hydrocarbon-air mixtures are independent, within the errors of measurement, of the proportion of the hydrocarbon within quite wide limits. For instance, when the proportion of heptane to air was varied in the ratio of 1 to 10, the ignition temperature was only lowered apparently by some 8° C. from 293° C. to 285° C. As the error of observation may certainly amount to 4° C. in such experiments, there is here no evidence that the alteration of the concentration of heptane has any effect on the ignition temperature. If, as might appear reasonable on the law of mass action, the rate of reaction depended directly on the concentration of the heptane, we should expect it to be 10 times as great in one case as in the other. Now, it is shown later in this paper that the temperature coefficient B, in the case of heptane, is about 13,000, from which it follows from equation (5) or (8) that if the initial rate of reaction on com- pression were really 10 times as great in one case as in the other, the ignition temperature of the richer mixture should be over 50° lower than that of the weaker mixture. One could not fail to detect with certainty a difference of this order : hence from the experimental results we come to the conclusion that the rate of combustion under these conditions does not depend on the concentration of the heptane vapour within wide limits. Experiments with other similar sub- stances support this view. YI. If this be correct, the only probable alternative is that the rate depends essentially on the concentration of the oxygen. We have not yet attempted to put this to a direct test in the apparatus used by us, since it would be necessary, to do this completely, to explode detonating mixtures of hydrocarbons and oxygen ; and we were anxious to avoid the danger of breaking the apparatus before other important experiments were carried out. We intend, however, to examine this point in the near future. It remains to consider whether, if the rate of reaction is directly proportional to the concentration of oxygen, the effect of the automatic decrease in oxygen content on ignition can be safely left out of account in calculating " delays " by * Tizard (he. oit.). Ignition of Gases by Sudden Compression. 97 means of equation (10). For this purpose we have to estimate to what extent the reaction occurs during the slow period of combustion before the explosion occurs, and the pressure- time curve becomes almost perpendicular to the time axis (see fig. 7). Taking the experiments on heptane as an example, we find (Table III.) that the biggest delay observed was about 0'6 second ; that the ignition temperature under these conditions is 250°C. = 553 absolute, and that the differ- ence of temperature between the gas and the walls was about 210° C. The observed cooling factor was 0*51. Since the specific heat of the mixture experimented with is approxi- mately = 0*2, the rate of reaction at the ignition temperature must correspond to a heat evolution of 0'51 X 0"2 x 240 = 24 calories per gram per second. The total heat of combustion of 1 gram of the mixture (containing about 75 per cent, of the theoretical amount of heptane for complete combustion) is about 510 calories, so that it is evident that during the period when the temperature is only rising slowly, which is always less than half a second, the amount of reaction and therefore the changes in con- centration must be small. Once the temperature begins to rise quickly, it is evident that the disappearance of oxygen can have only a secondary effect on the rate of the reaction compared with that of the rise of temperature until the com- bustion is nearly complete, so that the error involved in the calculation from equation (10) of the total time of combus- tion, by ignoring the effect of changing concentration, must be small. In view of the unavoidable experimental errors in carrying out experiments of this kind and of our still in- complete knowledge of the mechanism of combustion, we do not, in fact, think that any attempt to take fully into account such secondary effects is justified at present. VII. We therefore arrive at the conclusion that the time for complete combustion of an explosive mixture of gases when suddenly compressed to a temperature above its ignition temperature is closely given by the integration of equation (10), and is therefore t = — fh- - I at-^ d -(?} ' ' (11) 1 p 00 1 J-o /Jm ( > where T is the lowest ignition temperature under the con- dition of the experiment, and T is the initial compression temperature. Phil. May. Ser. 6. Vol. 44. No. 259. July 1922. H 98 Messrs. H. T. Tizard and D. R. Pye on the In this equation, -^ is given by equation (8); its value Wo depends on the magnitude of the temperature coefficient of the reaction. It is not possible to integrate the equation completely, but the integration can be carried out by approxi- mate methods for any value of ^- . In the table below are To given values of the expression a (-^n — )t a t different T l ° ' b values of ^ , and for various values of m . The i. ° i ° corresponding curves are shown in hg. 8. Fig. 8. If our views are correct, these should form standard curves representing the delays which should be observed under different conditions when any explosive mixture of gases is compressed to a temperature above its lowest ionition temperature. The application of the theory to any specific case should enable the temperature coefficient of the reaction to be determined. Ignition of Gases by Sudden Compression. 99 Table II. Values of a( ° \t for different values of r . r and . T/T . 6/T =12. = 10. =8. =6. = 4. 1004 •086 •112 •155 •233 •418 101 •056 •075 •108 •169 •321 103 •0245 •036 •056 •098 •209 107 •0081 •014 •025 •051 •131 1-11 •0033 •0065 •0135 •032 •095 1-27 •0002 •0007 I 1 •0022 •0083 1 •0415 The curves bring out clearly the effect of the two main factors which determine the characteristics of an explosion by sudden compression ; namely, the initial rate of loss of heat, and the temperature coefficient. If two gases have the same ignition temperature under the same conditions of loss of heat, the sharpest explosion will occur in the case of the gas with the. highest temperature coefficient, and the greater in this case will be the effect, on the magnitude of the delay before explosion, of: a higher temperature of compression. On the other hand, in any one case, the ignition temperature will be raised by carrying out the experiment under con- ditions which involve an increased rate of loss of heat ; at the same time the sharpness of the explosion will also be in- creased. VIII. To test the above theory, and to use it to obtain a measure of the temperature coefficient in certain cases, we chose three substances : heptane C 7 H 16 , ether C 2 H 5 .O.C 2 H 5 , and carbon bisulphide CJS 2 . These substances were chosen for the following reasons : (a) they could be obtained in a sufficiently pure state ; (b) they all have low ignition tem- peratures, which lessens the practical difficulties of the experiments ; (<•) they are known to behave very differently from the point of view of detonation when used as fuels for internal combustion engines; and (d) their difference in chemical and physical properties makes the comparison of their behaviour on combustion particularly interesting. To test the theory adequately we considered it absolutely necessary, particularly in view of the simplifying assump- tions made, not to be content with one set of conditions for the ignition experiments. Two series of experiments were therefore made with each substance; in the first series the H2 100 Messrs. H. T. Tizard and D. R. Pye on the gaseous mixtures with air were compressed in a non-turbulent condition, while in the second a high-speed fan was kept running throughout the period of compression and subsequent ignition. The use of the fan increased the rate of heat-loss at the compression temperature by about three times ; hence the difference in ignition temperatures observed with and without the fan running gives an important and necessary check on the value of the temperature coefficient which is calculated from the " delay " curve obtained when the non- turbulent gases are compressed. The temperatures given below represent the average temperature of the gas at the instant of maximum compression. By measuring the com- pression pressures in each experiment, a value of ei 7," the apparent ratio of the specific heats, is obtained from the expression : V •*- max. V where r is the compression ratio. The average value of 7 is taken for the series, and the compression temperatures then calculated for each case from the expression T y— 1 In each set of experiments the initial mixture of gases was of the same composition throughout, the proportion of air being somewhat greater than that required for complete combustion. The intention of using a weak mixture was to avoid as far as possible the deposition of carbon ; as stated above, the absolute ignition temperature is not affected appreciably by fairly wide changes in the original strength of the mixture. IX. The first results with heptane gave a very satisfactory confirmation of the theory developed above. The results of the experiments are given in the following tables and diagrams, which include measurements from all the records made under each set of conditions. No unsatisfactory records have been discarded. Ignition of Gases by Sudden Compression. 101 Table III. Ignition of mixtures of heptane and air. Mean apparent value of "7" observed = 1*313. Tan stationary. Initial pressure (atmospheric) == 11*8 lb./ sq. in. in expts. A x to A 10 , and 14'9 lb./sq. in. in expts. A u to A 17 . Strength of mixture = 1 grm. heptane : 20 grms. air. No. of Com pa. Initial Max. Compn. Max. Avge. temp, cale. from Delay Cooling factor expt. ratio. temp. pressure. y=l'313. obs. = a. A x 5'55 51° C. 140-5 lb./sq. in. 280° C. No ignition. 0-49 A., 603 50-5 156 295 019 sec. — A 3 702 50-5 192 323 0-04 — A, 8-02 49-5 225 346 0007 — Ag 5-55 46 141 273 No ignition. 0-53 A, 582 46 151-5 281 0-56 — A 7 6-23 46 163 293 0-21 . — A 8 8-02 46-5 227 340 Very small. — A 3 7 02 47 189-5 316 0-06 — A 10 702 42 ♦166 307 0-07 — A u 606 58 174 324 005 — A l2 (V56 53 177 315 006 — a" 6-56 48 176 306 0-12 — A 14 6-56 44 177 298 0-18 — A 1B 6-56 41 176 293 0-28 — 656 38 178 288 0-25 — A 17 656 35 176 282 0-58 The cooling factors in Table III. were obtained from the results of those experiments where no ignition occurred by the application of equation (3). The results for A x were as follows : — Table IV. Calculation of cooling factor. Time from max. pressure. Obs. pressure. lb./sq. in. P " P /- ^ F -T^- 1 - 8 140-5 58-5 023 sec. 134 52 ■0512 0-51 0-47 127-5 45-5 •109 0-53 0-7 123-5 41-5 •149 0-49 093 119 37 •199 0-49 M6 116 34 •236 0-47 1-39 112-5 30-5 •283 0-47 Mean = 0-49 P^ = 5*55 x 14-8 = 82-1 lb./sq. in. 102 Messrs. H. T. Tizard and D. R. Pye the The results for plate A 5 were similar, the mean being a = 0*53. In th© calculations the figure 0*51 has been taken. Table V. Ignition of mixtures of heptane and air. Fan full speed. Initial pressure = 14*86 lb./sq. in. Strength of mixture as before. Mean apparent value of 7 = 1*310. No. of Corupn. Initial Max. Compn. Max. Aver. Delay obs. Cooling expt. ratio. temp. pressure. temp. calc. factors. B x •• 6-03 61° C. 157*5 310° C. 16 sec. B .. 552 59 139 291 No ignition. 143 B, .. 5-73 59'5 148 298 }J 1-40 B 4 .. 591 59-5 154 304 55 1-36 B 5 .. 6-13 59-5 161 310 }J T47 B G .. 633 595 165 313 0-13 — b: .. 6-56 59 175 321 0-08 — B 8 .. 701 60 188 336 005 — B 9 .. 8-02 60 9 362 o-oo* — Mean= T42. * Ignited before top of compression. X. Considering firstly the results of the experiments without the fan running, we find that at the lowest ignition temperature of about 280° C, the temperature of the walls /HP A\ was about 40° C. The expression a ^— ^ (equation 11) T, has therefore the value — 0*51 x 240 553 0*22. Assume that the true ignition temperature under these conditions is 280° C. = 553 absolute, and that ^ -10: lo then by integration of the theoretical equation (see Table II.) we obtain the folio wing results : — Ignition of Gases by Sudden Compression. Table VI. Ignition of heptane by air. No fan. Lowest ignition temperature T taken as 553 absolute. 103 T assumed = 10. T/T . T. 1° C. a( ( \^)t(theov.). t (calc). 1-004 555 282 0-112 0-51 sec. 101 558*5 285-5 0075 0-34 „ 1-03 569-5 2965 0-036 01 64 „ 1-07 592 319 0-014 0063 „ 111 614 341 0-0065 003 ;, 1-27 702 429 0-0007 0-003 „ The last column contains the theoretical " delays " that should occur at compression temperatures given in the third column, if the temperature coefficient of the reaction corresponds to a value of ^ = 10. J-o Fig. 9 ft 350 ' I Heptane and A • + ^ ?""••— "*" N ° ' 9nlt '° n ' ^ V ^^ < K "* ~4- "*" No ignition. => + + 0-3 Seconds delay The corresponding theoretical curve is shown in fig. 9, the experimental points taken from Table III. being marked with a cross. The general agreement is all that could be desired. 104 Messrs. H. T. Tizard and D. R. Pye on the XL If the value for the temperature coefficient so deduced is correct, it should be possible to use it to calculate the higher ignition temperature when the fan is used, and also the shape of the new delay curve. Now, the mean cooling factor with the fan has been shown to be 1*4-2. The ignition temperature under these conditions is evidently about 310° C. (see Table V.), the temperature of the walls being 60° C. Hence the ratio of the rate of loss of heat with and without fan at the respective ignition temperatures is — Qo' _ 1'42 (310-60) Qo" = 2-90. 0-51(280-40) Hence the new (theoretical) ignition temperature T ' (with fan) should be given by (see equation (8)) — lo g ^ = lo g 2-90=|- o (l-|) = = 10 ( 1 -|) e. 0-462 or Jr = 0-954. T ' = 580 = 307° C. This is close to the observed value. tv, , J 6 .„ . 5530 Ine new value ot ™ will be "~^J7T 9' 5 ; while the value of (T o '-0\ 1*42x247 \ T ' ) = 0-60. T ' / ~ 580 The corresponding theoretical values of the delay are shown in the following table. Table VII. Ignition of heptane by air. Fan full speed. Ignition temperature calculated from previous results, 580° C. b_ To' = 9-5. T/T . T. *°C. « 1 m , ) t (theor.). t fcalc.). 1-004 582 309 012 0-20 sec. 1-01 586 313 008 0-13 1-03 597 324 0-04 0-07 1-07 621 348 0-016 0027 111 644 371 0-0076 0-013 Ignition of Gases by Sudden Compression. 105 The theoretical values given in this table are represented by the dotted curve in fig. 9, the experimental results (Table V.) being shown by circles. The close agreement between experiment and theory is obvious ; it is, indeed, closer than could reasonably have been expected in view of the fact that the temperature errors must be estimated as 3 or 4 degrees, while the cooling factors are subject to an error of about 5 per cent. The results can, however, leave little doubt of the substantial accuracy of the simple theory worked out above, and the temperature coefficient deduced must be very near the truth. It is of great interest to note that it is of the same order as that of chemical reactions in liquids at ordinary temperature ; for the reaction velocity is approximately doubled for a 3 per cent, rise in absolute temperature. XII. The only experimental values given in Tables III. and V. which seem to call for any special comment are those corresponding to experiments A 4 and A 8 . The " delays " found in these experiments were considerably smaller than those expected theoretically. This may be due to the fact that the measurement of very small delays is necessarily somewhat inaccurate with the apparatus used, since the speed of the piston falls off as the compression approaches its maximum. In such cases the lowest ignition temperature is, of course, reached before the piston reaches the top of compression, so that the measured " delays " which are measured from the time of maximum compression tend to be too small. But there is also a curious effect, which is invariably observed in these experiments on the self-ignition of carbon compounds, when the initial tem- perature is high, and the time of explosion short. It is always found that the explosion, though apparently sharp, is not complete, but that a fluffy deposit of carbon is thrown down. This deposition of carbon in an explosion has often been noticed by other workers when ignition is effected by a spark, but it is usually thought to be a consequence of having too little oxygen for complete combustion; in our experiments, however, the oxygen was always in considerable excess. When the minimum ignition temperature is not greatly exceeded, and when therefore the explosion is comparatively slow, combustion is complete, and no carbon deposit is formed. At higher initial temperature, however, one cannot escape the conclusion that the hydrogen is burnt preferentially to the carbon, and that the rate of combination of carbon atoms can be greater than the rate of combination 106 Messrs. H. T. Tizard and D. R. Pye on the o£ carbon with oxygen. The exact conditions when this occurs seems well worthy o£ further investigation. It is always necessary to clean out the cylinder carefully after such a deposit has been formed, and before the next experiment is made ; for, if not, abnormal results will be obtained, and the " delay " before ignition occurs will be very much shorter than is expected. To explain this it does not appear to be necessary to attribute any special " catalytic " activity to the carbon ; a simple physical ex- planation seems to be sufficient. Such a deposit is known to be a very bad conductor of heat. If left on the walls of the piston and cylinder, we shall therefore have, on the next compression, large portions of gas from which the heat cannot get away quickly. Hence the ignition temperature is lowered, and the explosion takes place more rapidly. XIII. Experiments on the self- ignition of mixtures of the vapour of ethyl ether C 2 H 5 .O.C 2 H 5 gave very similar results. The results of the experiments on the compression of non-turbulent mixtures are shown in Table VIII. ; while Tuble VIII. Ignition of mixtures of ether and air. Fan stationary. Initial pressure (atmospheric) =I4'77 lb./sq. in. Mean apparent value of "7" = 1*309. Strength of mixture = 1 part ether to 15 of air by weight. No. of Compn. Initial Max. Compn. Max. temp. Delay Cooling expt. ratio. temp. pressure. 7 = 1-309. obs. -a, Oi 451 25° C. 105 lb./sq. in. 201° C. No. ign. 0-47 c 2 4-83 24 116 211 ,, 0-47 °3 502 23 121-5 „ 214 041 — O! 5-21 23 128 „ 220 030 — C 5 5-42 23 135 - 5 ,, 226 0-205 — C 6 5-63 23 143 232 012 . — c~ 5-82 23 149 237 0093 — Cs 6-03 23 155 242-5 0071 — c 9 0-56 23 172 256 035 — Table X. shows the results when the fan was rotating at full speed. No ignition was observed in experiments Qi and C 2 in the first series, and in experiments D l5 D 2 , and D 7 in the second series. The details of the calculation of the cooling factor corresponding to experiment C 2 are shown in Table IX. The cooling curve was not so regular in the case of Cj, but the mean value of the constant was the same, Ignition of Gases by Sudden Compression, Table IX. ( 1 alculation of cooling- factor corresponding to C 2 . 107 Time from maximum pressure. Observed pressure. P-P r , ■*• max. — •*- f j •=2-8 a/*. 00 116-2 44-9 0-232 sec. 111-7 406 •044 •44 •465 ., 107-3 36-0 •096 •47 •70 „ 103- > 323 •143 •47 •93 „ 1006 29-3 •185 •46 1-16 „ 96-9 256 •244 •48 1-39 „ 94-7 23-4 •283 •47 1-63 „ 92-5 21-2 •326 •46 i p / =4-83x14 77=71-3. Mear = 0-47 Table X. Ignition of mixtures of ether and air. Fan full speed. Initial pressure (atmospheric) = 14*77 lb. /sq. in. Mean apparent value of y=1'308. Strength of mixture as in Table VIII. No. of Compn. Initial Max. Couip. Max. temp, calc. from Delay Cooling factor expt. ratio. temp. pressure. y = 1'308. obs. =«. Di 5-42 24° C. 135 226-5° C. No. ign. 1-33 D 2 573 24 146-5 235 1-28 D 3 6-03 24 155-5 243 5 o-io — vl 633 24 164 251 047 D 5 6-63 24 174 259 0-038 — Dc 7-02 24 186 268-5 0-026 Dt 5-82 22 149 235 No. igu. 1-25 D fl 5-94 22 150-5 238 013 — D g 6-03 22 155 240 0-087 _ Dio 6-13 22 161 243 0-078 — Mean =1-29 namely a =0*47. Table XI. shows the details of the cal- culations for D 2 ; it will be noticed that the cooling factor appears to diminish fairly steadily as the time increases. T>i and D 7 showed a similar effect : in each case the mean has been taken, and the mean value for the three experi- ments, namely a = 1'29, is probably fairly accurate. The 108 Messrs. H. T. Tizard and D. R. Pye on the values for the cooling factors so obtained are lower than those found in the heptane experiments ; this would be expected, for, although the specific heats of the mixtures are about the same, the compression ratios used were lower in the case of ether mixture than in the case of heptane, since ether has a considerably lower ignition temperature. Table XI. Calculation of cooling factor corresponding to D 2 . Time from maximum pressure. Observed pressure. P - P /- p p , x max. *- f * log r-p, - =e - i a =2-3 S/t.\ 146-5 lb./sq. in. 61-8 •232 sec. 129-5 44-8 •140 . 1-39 •465 „ 118-4 337 •263 1-30 •70 „ 109-5 25-8 •379 1-25 •93 „ 103-6 19-9 •492 1-22 1*16 „ 97-7 130 •677 1-34 1'39 „ 96-2 11-5 •730 1-21 P f =5'73x 14-77=84-7 lb./sq. in. Mean = 1-28 XIV. Taking the ignition temperature T (without fan) as 212° C. = 485 abs., the experimental observations of the delay before ignition at higher temperatures are reproduced closely if the value of ^ =11. In this case, we have T -e _._ /212-23\ : 0-183. The theoretical values for "t" calculated according to equation (11), are shown in the following table. Table XII. T/T . T. | t°C. 1-004 487 214 1-01 490 217 1-03 499-5 226-5 1-07 519 246 1-11 538-5 265-5 127 616 343 a(i^)^(theor.). •098 •065 •030 •0105 •0046 •0004 t (calc.). )535 sec. •355 •164 •057 •025 •002 Ignition of Gases by Sudden Compression. 109 The lower curve in fig. 10 is the theoretical curve derived in this way ; the experimental points are shown by crosses. In the series of experiments with the fan running at full speed, "a" = 1*29, and the new ignition temperature T</ is approximately 235° 0.= 508 abs. Fig-. 10. Ethyl ether and air. V* Nfcf^ ^Nfci *~~ 2e*pts.) -f- 1 No ianitior and 2 03 04 5 Seconds delay. Hence .( W) = 1-29 x (*«) =0-54, Qo'_ 1-29 (235-23) _ Qo "0-47(212-23) °* ., b Hence if fit =11, T ' should be given by log3-08 = 0-489 = ll(l-|j-,), or ^=0-956; .-. T ' = 507-5 (abs.), which is evidently very close to the observed value. Taking this value for T ', we have b in 485 1/w , T7 = 11X 507-5 =1 ° ,D - 110 Messrs. H. T. Tizard and D. R. Pye on the Table XIII. gives the theoretical values for the time of ignition at various temperatures, calculated as already V11U&U ) l " 1Y111 5 ^0 - J-o Table XIII. T/T . 1-004 101 1-03 1-07 111 T. t°c. a( m , U(theor.). t (calc). 509-5 512-5 522-5 543 563-5 236-5 239-5 249-5 270' 290-5 0-104 0-069 0-0325 0-012 0-0055 0-19 sec. 0-13 „ 06 „ 0-022 „ o-oio„ The corresponding curve is the upper dotted curve in fig. 10, the experimental values being shown by circles. The general agreement is again all that could be desired. Although the ignition temperature of ether is very much lower than that of heptane, the temperature coefficient of the combustion reaction is the same within the experimental errors involved. XV. The experiments on carbon bisulphide were expected to be of particular interest, on account of the" anomalous behaviour of this substance if used as a fuel in internal combustion engines. It is known that for any given fuel the highest thermal efficiency obtainable is limited mainly by the tendency to " detonation " at high compression ratios. It is usually assumed that the tendency of any fuel to detonate depends upon its ignition temperature ; the lower the ignition temperature, the greater will be the tendency to detonation. According to our views, the ignition temperature is not a safe criterion of the tendency to detonation ; the temperature coefficient is also an important factor which must be taken into account The use of carbon bisulphide as a fuel illustrates this point very well ; although it has a lower ignition temperature than heptane, it detonates less easily in internal combustion engines, and not more easily, as might be supposed. We expected, therefore, to find, by the experiments described in this paper, that the temperature coefficient of its reaction with oxygen was very distinctly lower than that of heptane and similar substances. The experiments fully confirmed this, although the results do not appear to be so satisfactory in all respects as those carried out with heptane and ether. Ignition of Gases by Sudden Compression. Ill Tables XIV. and XV. contain all the experimental results of the ignition at various temperatures of mixtures of CS 2 and air (a) when initially stagnant, and (b) when the fan was rotating at full speed. Table XIV. Ignition of mixtures of CS 2 and air. Fan stationary. Initial pressure (atmospheric) =• 14*60 lb./sq. in. Mean apparent value of u y" = 1*332. Strength of mixture = 1 part CS^ to 8 parts air by weight. No. of Compn. Initial Max. Compn. Max. temp. Delay Cooling exp. ratio. temp. pressure. y = 1-332. obs. factor. Ei ... 603 48-5° C. 159 lb./sq. in. 313° C. 00S6 E, ... 5-63 47*5 143 „ 296 0-115 E 3 ... 5-63 42 143 286 0-18 E 4 ... 563 39 H7 281 0-26 Eg .. 5-63 36 147 275-5 0-35 E(j ... 5-63 32 145 268-5 0-42 1 E- ... 5-02 32 1265 „ 248 No ign. 0-46 E* ... 5-21 315 1325 „ 254 0-71 E 9 ... 5-42 31-5 140-5 „ 261 0-59 Eie... 5-82 31-5 154-5 „ 273-5 034 En- 7-02 31 193 308 0-087 Table XV. Ignition of mixtures of CS 2 and air. Fan full speed. Initial pressure (atmospheric) = 14V6 lb./sq. in. expts. Fi-F 6 . = 14-0 Mean apparent value of 7 = 1*323. Strength of mixture as before. Fr-F, No. of Compn. Initial expt. ratio. temp. , Ei ... 633 43°C. F 2 ... 6-44 43 F 3 ... 6-56 43 :f, ... 663 42-5 F« ... 6-84 43 F 6 ... 7*55 43 F 7 ... 6-56 43 F, ... 7-02 43 *Fn ... 8-02 43 F 10 ... 10-0 42 Max. Compn. pressure. 169 lb 171 177 178 185 208 175 191 224 ? /sq.in. Max. temp. calc. from 7=1-323. 300° 304 307 309 315 334 307 320 346 391 Delay Cooling obs. factor. No ign. i-3o ; 014 — Noign.(?) 1-38 0-13 1 012 — 005 — 011 — 0-09 — 003 — Ignit'd bc- fon top of compn. Mom I -34 The cooling curves were very uniform in the above cases. 112 Messrs. H. T. Tizard and D. R. Pye on the XVI. It is clear, from the shape of the curve connect- ing time of ignition with temperature, from the results of experiments without the fan, and from the difference in ignition temperature observed with and without the fan, that the temperature coefficient of the reaction is low. If we take T , the ignition temperature without the fan, to be 253° 0. = 526 abs., and ^ = 7, we obtain the theoretical values for the time of ignition from equation (11) which are given in Table XVI. The value of o^~~ e is 0*46 x 25 ^~ 32 = 0*193. 1 526 Table XVI. T = 526abs. Z>/T = 7*0. T/T . T (abs.). t°G. a(?^)*(theor.). t (calc). 1-004 5228 255 0-187 •98 sec. 1-01 531 258 0-133 •70 „ 103 542 269 0-072 •38 „ 1-07 563 290 036 •19 „ 111 584 311 0-020 •105 „ 1-27 66S 395 0-004 021 „ Fig. 11. 1 it Carbon disulphic e and a [N S. T.rr e scale ^ alf that of Figs. 10 and 1 No ionit "■] + ^ + + No ignition. 0-2 0-3 04 0-5 0-6 07 OS 09 Seconds delay. The theoretical values are shown by the lower curve in no-. 11. Ignition of Gases by Sudden Compression. 113 The cooling factor when the fan is used is 1*34. The ratio of the loss of heat (or reaction velocity) at the ignition temperatures with and without fan is consequently Qo' 1-34(296-42) Qo ~ 046(253-32) = 3-35. Hence the new ignition temperature T ' should be given by the expression log3-35=0-525=£/T (l- ^ ■(•-&)■ or ^ =1-0-075 = 0-925, T ' = 568-5 = 295°'5C. The new value of rfr7 corresponding to this is therefore J-o v-'-(«w-** The calculation of the corresponding delay curve is shown ° T , I being l-34xg=06. T/T . 1004 101 1-03 107 1-11 571 574 585-5 6085 631 Table XVII. t°c. 298 301 312-5 335-5 358 a(^rr^(theor.). 0205 015 0-08 0042 0-025 t (calc). '34 sec. •25 „ •13 „ •07 „ •04 „ This is the dotted curve shown in fig. 11, the experimental points being marked as before. The agreement between Phil May. S. 6. Vol. 44. No. 259. July 1922. I 114 Messrs. H. T. Tizard and D. R. Pye on the theory and experiment is in this case only moderate. In particular, it will be noticed that although the ignition temperature with the fan running is calculated, from the results without the fan, to be 295°*5 C, actually no ignition was observed at 300°C, and even in one experiment at 307° C, although this is extremely doubtful, since in two other experiments ignition was observed to take place at 307° C. and 304° C, with the comparatively short delays of O'll and 0'14 second respectively. It is possible that when the loss of heat is considerable, and the temperature coefficient small, there is an appreciable error introduced in neglecting changes of concentration when calculating the time of ignition at temperatures near the ignition temperature. This would account for no ignition being observed when a long delay was expected. The value of 6/T given above cannot, bow- ever, be very far wrong. For suppose we take the ignition temperature T ' of the mixture when, the fan is running as 300° C. instead of 295 0, 5, and calculate the temperature co- efficient solely from the difference in T and T ' without regard to the " delay " curve when the fan is stationary ; then we shall have Ti=||6 =0 .9i8 lo 573 and Qo'_ 1-34( 300-42) _ Qo ~ 0-46 (253 - 32) ~ 6 W ' log3-40 = -531=A(l-g,) = |- x-082; J-O — =6*5 instead of 7*0. J-o This value for ~r would, in fact, fit the lower part of the J-o delay curve without the fan rather better than the value 7*0, but the calculated delay curve when the fan is running- would then be some way from the experimental points. If we take a mean value 6/T =6-7, we shall be very unlikely to be as much as 10 per cent, from Ignition of Gases by Sudden Compression. 115 the true value, even when all possible sources of error are taken fully into account. XVII. It has already been mentioned that when the time of ignition by compression of hydrocarbons (and of ether) is small, i. e. when the gases are suddenly compressed to a temperature well above the ignition temperature, carbon is thrown down, even though excess of oxygen is present. In the experiments with CS 2 an even more curious phenomenon was noticed. In this case, whereas the sulphur burns to S0 2 when the initial temperature of compression does not exceed very greatly the lowest ignition temperature, when the initial temperature is high the products of combustion smell strongly of JBLS. For instance, the products of com- bustion in experiments E x and E n above both smelt strongly of H 2 S, although in E 2 -E ]0 inclusive only S0 2 could be detected by smell. It was also possible to detect H 2 S after experiments F 6 and F 10 (with the fan), the smell being particularly strong in the case of F 10 . The smell of H 2 S could also just be detected along with S0 2 in experiment F 9 , whereas in the remainder only S0 2 could be detected. The H 2 S could only have come by combination with w r ater- vapour present in the air, which was not dried. This occurrence of H 2 S is all the more interesting since it is known that a perfectly dry mixture of CS 2 and oxygen can be exploded by a spark, whereas perfectly dry mixtures of other gases, e. g. carbon monoxide with oxygen, cannot. It is possible that some such reaction as CS 3 + 2H s O = CO s + 2H s S takes place, followed by the combustion of H 2 S ; but even if this is the case, it would be expected that the H 2 S would be quickly burnt in presence of excess of oxygen under the conditions of these experiments. Further experiments on the ignition of H 2 S itself will probably throw some light on these observations. XVIII. In Table XVIII. are summarized the chief results of the experiments described above. The ignition tempera- tures represent the lowest average temperatures at which the non-turbulent mixture could be caused to ignite. The rates of evolution of heat at these temperatures for the three cases are calculated from the cooling factors and the specific heats of the mixtures. 12 116 Messrs. IL T. Tizard and D. R. Pye on the Table XVIII. Heptane C 7 H 16 . Ether C 2 H 5 .O.C 2 H 5 . Carbon bisulphide cs 2 . Composition of gas by weight... 1 : 20 of air. i 1:15 1:8 T =igni(ion temperature " 280° 0. 212° 0. 253° C. c v at ignition temperature ' 0'20 0'20 0-18 Rate of evolution of beat due to; 25 calories . ~ reaction per gram of mixture per second. [ 18-5 Total beat of combustion pei 1 510 calories. 510 calories. 386 calories. Value of b/T ■ io-o+5 % ! n-o+5°/_ 6-7 ±10% XIX. In order to calculate the true temperature co- efficient B (see equation 5) from the values of b/T , it is necessary to examine the significance of T a little more closely. As already stated, T is a measure of the lowest average temperature of the gas at which ignition takes place. Now the actual temperature of the gas after sudden compression can hardly be uniform throughout ; in fact, when the gas ignites after a considerable delay, it is always found that the pressure, and therefore the average temper- ature, falls, in some cases quite considerably, before ignition takes place throughout the mass. This shows clearly that that portion of the gas which ignites at first has initially a higher temperature than the average, thus confirming Dixon's experiments. Absence of information as to the temperature gradients which may exist under these conditions has no doubt led Nernst and Dixon in their experiments to calculate the ignition temperature as if the compression were adiabatic, and to ignore the influence of loss of heat during compression and before ignition. They assume, in fact, that that portion of the gas which does ignite is at the adiabatic temperature. It is hardly likely, however, that big differences in temperature exist after compression when the gases are in a turbulent state ; and the fact that the temperature coefficients, calculated from the differences in "average" ignition temperature between turbulent and non-turbulent mixture, Ignition of Gases by Sudden Compression. 117 agree so well with those calculated from measurements of time of ignition at various temperatures with non-turbulent mixture, confirms the views taken in the previous paper (Tizard, loc. cit.) that it is unlikely also that any big differences of temperature exist in the non-turbulent mixture after compression. In the absence of direct evidence on this point, however, it is important to calculate the " adia- batic " temperatures also in the above cases. The mean specific heats C„ per gram molecule are : Heptane (room temperature-300°C.) = 50 calories approx. Ether (65°-230° C.) = 33'6 calories (Regnault), CS 2 (70°-194° O.)=10'0 (Regnault). Taking these values, and C y = 5'0 for air, we obtain the figures for the mean true value of " y" and the corresponding adiabatic temperatures, given in the following table. Table XIX. Heptane. Ether. OS 2 . Mean apparent value of " y " (observed) Mean true value of y ... " Average" ignition teni- 1-313 1-353 553 abs. 1-309 1-347 485 abs. 516 31°C.=6-4°/ 1-332 1-384 526 abs. 572 46° 0. =8-7% il Adiabatic" ignition 591 41° C. =7-4% The average specific heat for CJS 2 taken in the above calculations is probably too low, since it refers only to a range of temperature up to 194° C, whereas the ignition temperature was 260° C. It will be observed that the difference between the average observed and the theoretical adiabatic temperatures is not very great ; we consider that the "average" temperature is probably closer to the true ignition temperature than is the "adiabatic" temperature, but for the purpose of estimating every possible source of error in the temperature coefficients, it is better at this stage to recognize the uncertainty, and take for the true values of the ignition temperatures the values Heptane 573° absolute Ether 500° „ V ±4 per cent. CS 9 549° „ } 118 Messrs. H. T. Tizard and D. R. Pye on the Hence, from the values of =^ we get finally for the true temperature coefficient B : ° Temperature coefficient B (equation 5). Heptane-air... 13,200+ 9 per cent. Ether- „ ... 12,600 ± 9 „ 0S 2 - „... 8,500 + 14 „ The significance of these figures will perhaps be better appreciated by the statement, that the percentage rise in absolute temperature necessary to treble the reaction velocity is 4 per cent, in the case of heptane and ether and 7 per cent, in the case of CS 2 . XX. Of recent years, considerable attention has been directed to the "radiation" theory of chemical reactions. According to this theory, the ultimate cause of any chemical reaction is to be found in the absorption of radiation of a frequency which depends upon the nature of the reactants. In the case of the majority of chemical reactions, namely those which are not " photochemical " in nature, this radiation will belong either to the visible, or more usually in the short infra-red part of the spectrum. The supporters of the theory hold the view that it is only through the absorption of such radiation that a molecule is able to acquire that excess of energy, over the average at any temperature, which enables it to decompose or to react with another molecule. The frequency of the radiation is therefore known as the "activating" frequency. This reasoning leads to the conclusion that the temperature coefficient B of a mono-molecular reaction is determined by the relation B-£, d2) where "»>" is the activating frequency, which should correspond to an absorption band in the reacting species. No reliable experimental evidence has yet been brought forward in support of this theory, but in view of the scanti- ness of the data existing on homogeneous gas reactions, it is of particular interest to apply it to the results of the experiments described above. In attempting to apply the theory, a difficulty at once arises. Evidence has been brought forward to show that the ignition temperature of substances with oxygen is Ignition of Gases by Sudden Compression. 119 practically independent of the concentration of the com- bustible substance. If the rate of the reaction were deter- mined solely by the amount of oxygen present, we might expect the temperature coefficient also to depend solely on the oxygen, and therefore to be the same in all cases. This is clearly not true. Nor does the temperature coefficient, in the case of the heptane explosion which has been most closely investigated, correspond at all closely to that calculated by means of equation (12) from the infra-red absorption of oxygen. Oxygen has an absorption band corresponding to \ = &2fi, or j/ = '94x10 14 ; hence, since \ =4*86 x 10" 11 , we should have * B = 4550 (calc.) instead of 13,200 + 9 per cent. (obs.). It is clear that equation (12) cannot be applied. On the other hand, if the rate of reaction depended on the product of the number of active molecules, both of oxygen and the other reactant, we should expect, on the same theory, to find the temperature coefficient given by B = yO'i + 'a), where v u v 2 correspond to absorption bands in the reacting substances. Now, all hydrocarbons have a weak absorption band at \ = 24 /x, and a fairly strong one at 3*43 /jl. Taking \ = 2*4 fi, which is most favourable to the theory, we have v 2 = 1-25 x 10 14 and Vl (oxygen) ='94 x 10 14 . Hence B = 4-86 x 10" 11 x 2'2 x 10 14 = 10,700. This approaches more closely the experimental value B = 13,200 ±9 percent. It must be pointed out, however, that this approximate agreement is only obtained by an assumption as to the actual mechanism of the reaction which does not agree with the existing experimental results. The failure of the " radiation theory " to account for the results obtained in these experiments is more significant when we regard it in a different way. The theory requires that the rate of a chemical reaction should be proportional to the density in the reacting system of the radiation which is absorbed by the reacting substances. Now, in the case of gases which are caused to react by a rise in temperature due to sudden compression, the radiation density must remain practically unchanged, for the temperature of the walls remains constant. It may be momentarily increased 120 Messrs. H. T. Tizard and D. R. Pye on the owing to the sudden compression, but such an increase cannot persist during the period of delay, and in any case is negligible compared with the increase in radiation density which would occur if the temperature of the walls of the vessel were raised to the compression temperature of the gas. Again, the emission and absorption of radiation by the gas itself at the compression temperature of 500-600 absolute is negligible compared with that of the solid walls. Hence we arrive at the conclusion that, although the density of radiation in the system is not appreciably changed, the gases react at a high rate. This fact appears to us to prove conclusively that the radiation theory cannot be accepted either in its original form, or as modified to meet its failure to account quantitatively for the temperature coefficients of chemical reactions in liquids under steady conditions of temperature. It must be pointed out, however, that in spite of this con- clusion, there does seem to be an indirect connexion between the effect of temperature on the rate of combustion of many substances and their absorption of infra-red radiation. For example, Ooblentz has shown that all paraffin hydrocarbons have very similar absorption spectra, with a weak band at \ = about 2*4 //,, and strong bands at X = 3*43, 6*86 yu,, etc. Now, Bicardo has shown that the tendency of hydrocarbon fuels to detonate in an internal combustion engine depends consistently on their ignition temperatures as determined in the manner described above. According to our views this is strong evidence that the temperature coefficients are practically the same throughout. Again, it has been shown that ethyl ether has approximately the same temperature coefficient of combustion as that of heptane; while Coblentz has found that it's absorption spectrum is also nearly identical, with bands at 2*4 p and 3'45 /x. Carbon bisulphide, on the other hand, has a much smaller tendency to detonate in an internal combustion engine than heptane, although it has a lower ignition temperature; corresponding to this we find that the temperature coefficient is low, and that the first strong absorption band in the infra-red occurs at A, = 4* 6 //,. Finally, hydrogen " detonates " easily in spite of its high ignition temperature ; its temperature coefficient must therefore also be high, a deduction which is confirmed by some preliminary experiments we have made on the delay before the ignition of a non-turbulent mixture of hydrogen and air. We should expect from this point of view to find an absorption band in the short infra-red region (say about 1*0 yu,); actually no absorption is observed, but that the Ignition of Gases by Sudden Compression. 121 frequency of atomic oscillation is high is in agreement with our general knowledge of the hydrogen molecule. In spite, therefore, of the strong arguments that have been brought forward against the radiation theory of chemical reactions, these results support the view that there is a connexion, even though an indirect one, between the temperature co- efficients of gaseous reactions and the infra-red spectra of the reacting substances. XXI. The results of this investigation may be summarized as follows : — (a) Quantitative experiments confirm the view that at the lowest ignition temperature the heat evolved by the combustion of a gas just exceeds that lost to the surroundings. (6) From measurements of the rate of loss of heat just below the ignition temperature, and of the intervals between the end of compression and the occurrence of ignition at different temperatures, it is possible to deduce the temperature coefficient of the gaseous reaction. (c) The temperature coefficients so obtained are confirmed by the increase in the minimum ignition temperature which is observed when the gas is in a turbulent state. (d) The results show that the temperature coefficient of the combustion of carbon bisulphide is much lower than that of heptane or ether. This is in agreement with the relative tendencies of these fuels to detonate in an internal combustion engine. (e) The results do not agree with the radiation theory of chemical reaction. (/) Some evidence is put forward to show that the rate of reaction on sudden compression is independent within wide limits of the concentration of the com- bustible gas, but only depends on the amount of oxygen present. This evidence is, however, incom- plete. We are greatly indebted to Messrs. Ricardo & Co. for the loan of their apparatus and for much additional assistance ; also to the Department of Scientific and Industrial Research for a grant towards the expenses of the investigation. We also take this opportunity of thanking Mr. C. T. Travers for his help in carrying out some of the experiments. [ 122 ] IX. On the Vibration and Critical Speeds of Rotors. By C. Ropgeks, O.B.E., B.Sc, B.Eng., M.I.EE* NUMEROUS papers have been written on the question of the whirling and vibration of loaded shafts and kindred subjects, and the calculation of the first critical speed — the lowest speed at which the vibration shows a maximum value, is now a matter of daily routine in designing offices. This critical speed can be calculated with sufficient accuracy for practical purposes and as a rule the running at speeds not in the neighbourhood of that indicated by the calculation is free from vibration. But cases occasionally arise where troubles from vibration occur at speeds above or below the calculated critical speed, the reason for which is obscure and the remedy correspondingly difficult to find. It is the object of this paper to discuss various subsidiary causes which might conceivably lead to unsatisfactory run- ning at other than the usual calculated critical speed, but while these are indicated as possible causes of disturbance, it is not to be assumed that these causes always exist or that they will always induce disturbed running. The object is rather to indicate reasons why vibration might possibly arise and thus if an actual case occurs, to suggest a clue to the cause. Although the basis of the paper is a physical or mechanical one, the treatment is largely mathematical, as it is only by this means that formulae can be obtained from which numerical results can be worked out. The phenomena when a rotor vibrates are complicated, as the shaft is supported in the bearings on a film of oil, the thickness of which is continually changing, the bearings and foundations are not themselves perfectly rigid, and there is a certain amount of initial bending of the shaft (and to a much smaller extent of the rotor body) due to gravity. If the rotor consists of a number of disks as in a steam turbine, there is also the inter-action of the forces of each disk on the others. For the sake of simplicity, we shall confine our attention to the case of a single part rotor, either a disk or a cylinder, rigid as regards bending and mounted on an elastic shaft running in rigid bearings. Some effects of non-rigidity of the rotor and bearings and of alterations in the thickness of the oil film in the bearings will be indicated. * Communicated by the Author. On the Vibration and Critical Speeds of Rotors. 123 A single part rotor can vibrate in either of two ways, as shown in figures 1 and 2, or in a manner which is a combi- nation of the two motions : — Fig. 1. Fig-. 2. Fig. 1 shows a purely transverse vibration, while in fig. 2 the motion is solely one of oscillation about the centre of gravity. In the transverse vibration the conditions are clearly the same whether the rotor body is a disk or is cylindrical ; but in the case of the oscillation, the motion, owing to the gyrostatic effects called into play, depends both on the proportions of the rotor and the speed at which it is running. The speed at which transverse vibration becomes a maxi- mum we shall call the "first critical speed/' and that at which the oscillation becomes a maximum, the "second critical speed," as the latter is in all practical cases con- siderably higher than the former. The following is a general outline of the treatment adopted and the conclusions reached : — Section 1 deals with the vibration of a rotor when not running, and a relationship is deduced between the fre- quencies for the transverse motion and for the oscillation which we shall call respectively the " stationary first critical speed," and the " stationary second critical speeds." The second section deals with the transverse vibration, frictional resistance being ignored. It is first showm that there appears to be no foundation for the frequently ex- pressed view that there is a possible region of marked vibration at — ^ times the first critical speed, as such a conclusion can only be reached through an incorrect assump- tion with regard to the conditions. It is then shown that the motion or vibration is a circular whirl about the statically deflected position of the shaft, and that this motion reaches a maximum at a speed equal to the stationary first critical speed. The magnitude of the whirl is proportional to the amount by w'hich the machine is out of balance, so that the main vibration here dealt with should disappear with good balancing. The action of gravity is then gone into more fully, and it 124 Mr. 0. Rodger s on the Vibration is shown that in addition to producing the ordinary static deflexion, the action o£ gravity is such as to cause a double frequency ripple in the whirl which would tend to reach a maximum at half the first critical speed. The magnitude of this ripple is, however, proportional to the square of the amount by which the rotor is out of balance, and would therefore fail to appear in a well-balanced machine. In any case the effect is very small. It is then shown that a rotor with bi-polar asymmetry, such as exists in a rotor slotted for a two-pole winding, may show a double frequency vibration at half the critical speed even when the rotor is perfectly balanced, so that such a machine might vibrate at half the critical speed even when it would run perfectly at the full critical speed. Vibration arising from this cause could not, therefore, be rectified by balancing, and this is the only case met with where improved balancing would not effect an improvement in the running. This case is gone into in some detail, and it is shown that the motion here also is a circular whirl of double frequency, that is, of twice the speed of rotation of the machine. If, in addition, the machine is out of balance, a triple frequency effect might appear, but is not likely to do so. The effect is then discussed of lack of proportionality in the deflexion of the shaft and again the possibility appears of vibration appearing at half the critical speed, but only if the machine is not properly balanced. The effect is then gone into of fluctuations in the angular velocity through variations in the driving torque, and of resonance between the rotor and the foundations or other masses outside the machine, from which it appears that marked vibration might appear at almost any speed through either of these causes. The effect of friction on ihe transverse vibration is then discussed, and the results are given for the case where the frictional resistance varies as the first power of the speed, and also where it varies as the second power of the speed, the latter being more probably in accordance with the facts than the former. It is shown that the maximum vibration appears in both cases at a speed equal to the stationary critical speed, also that the phase difference between the force due to the out-of-balance and the displacement depends on the amount of friction, and also on the speed. If the frictional forces vary as the square of the speed, as is probably the case, the angle varies also with the amount by which the rotor is out-of-balance. Some effects of viscosity of oil in the bearings, and of different bearing clearances are then gone into. 7?r and Critical Speeds of Rotors. 125 In Section 3, the oscillatory vibration is dealt with, taking into account the gyrostatic effects when the machine is rotating, but ignoring the friction in order to keep the expressions as simple as possible. It is there shown that the gyrostatic effect causes the point of marked vibration to occur at a higher speed than would be the case if the machine were not rotating, and simple rules are given for calculating this vibrating speed. An example is added to illustrate the method of working the rules given. Much of the work on the main transverse vibration and the main oscillatory vibration has been dealt with in various forms by Ghree, Stodola, Morley and others, and the solution for the transverse vibration with friction depending on the first power of the speed has been given by H. H. Jeffcott (Phil. Mag. March 1919), but the ground covered by the remainder of the paper, particularly the question of sub- sidiary critical speeds, does not appear to have received much attention ; there is, however, in 'Engineering' a dis- cussion where subsidiary critical speeds are touched on, arising out of a paper by W. Kerr in that journal (Feb. 18th, 1916). Section I. — Stationary Vibrations. A. Transverse Vibrations. 1. If M is the mass of the rotor body (the mass of the shaft being being neglected), and we assume the rotor to be perfectly balanced, the shaft will, when not rotating, show a deflexion measured at the centre of gravity of the rotor of where a is the force required to produce unit deflexion. The method of working out the static deflexion of the rotor for actual cases is well understood and the value of a can be got from the deflexion diagram. 2. If now the rotor is set in vibration in a vertical plane, the motion is represented by the following equation (using d 2 y fluxional notation, where y is written for -~^ and y for 7 at ^,etc.) ■ ' My + <ry + M ff = (2) 126 Mr. 0. Rodgers on the Vibration The solution is ^MVl^y^ ■ ■ ■ m where Nx and y ± are constants the values o£ which depend on the initial conditions. The vibration therefore takes place about the statically deflected position as a centre, and with a frequency of vibration of — c l9 where c '=Vm w This vibration takes place in a vertical plane and may be considered as the resultant of two vectors rotating in oppo- site directions, each with an angular velocity of \/ ^ . If o- and M are expressed in eg s. or f.p.s. units, this angular velocity will be in radians per second and since from (1), ^ is numerically equal to -^ , the speed of either of these vectors in revs, per minute will be _: — A / — . If, further 2-7T V y Q g and y are in c.g.s. units we have the formula WTJ ,, 60 /Ml 300 . , . /K . where y is the static deflexion in cm. 3. It will be seen afterwards that, as is well known, this formula gives the first critical speed in R.P.M. ; this is to be expected, as the out-of-balance forces will then resonate with the natural free vibrations, with the result that the latter will become of considerable magnitude. B. Oscillatory Vibrations. 1. If the rotor is twisted about its centre of gravity so that the deflexion is in a vertical plane, and is allowed to oscillate freely, the motion is represented by B^ + /c^ = 0, (6) where B is the cross moment of inertia, that is, the moment of inertia of the rotor about a line through the centre of gravity at right angles to the shaft, yjr is the angle through and Critical Speeds of Rotors. 127 which the axis of the rotor at its centre of gravity is deflected from the stationary position, and k is the torque required to produce unit angular deflexion. The solution is ^ = N 2 sin(^/^- 72 ^ .... (7) where N 2 and y 2 are constants the values of which depend on the initial conditions. The frequency of the oscillation is therefore iiv b or L c " where '-a/b («) As already indicated, we cannot at once deduce from this what will be the actual second critical speed, owing to the gyrostatic effects, but the result is of importance, as it simplifies the calculation of the actual second critical speed, as will be shown later. We shall in what follows call c 2 the stationary second critical speed. 2. It should be pointed out that there is a simple relation between c x and c 2 which greatly facilitates the calculation of the stationary second critical speed in those cases where the centre of gravity of the rotor is midway between the bearings. If 21 is the distance between the bearing centres and P 2 the force exerted by the deflected shaft on either bearing, /n/r = 2P 2 Z. The angle y]r is very small so that the force P 2 is the same as would be required to depress the shaft through a distance -tyl if the rotor were held rigidly. Now we have seen that the force M</ at the centre of gravity causes a transverse deflexion of ?/ = , and as yfr is small, so that Mg — ayfrl, also /n/r = 2P 2 Z and F 2 = ±Mg ; therefore fcyjr = MpZ, so that K = a-l 2 (9) and (8) becomes 7 / a .■ C2==/ V B' ^ ' 128 Mr, 0. Kodgers on the Vibration and if B = M^ 2 2 , c ^h \/w ~N (12) and comparing with (4) we thus get : C\ Aug c 2 ~~ I We thus find that First critical speed (transverse vibration) Stationary Second critical speed (oscillation) _ Radius of Gryration for the cross moment of inertia Half the distance between the bearing centres This is a useful formula for calculating the stationary second critical speed when the first is known, for cases where the centre of gravity is midway between the bearings. It shows that with cylindrical rotors of this type the second critical speed must always be considerably above the first, and the only instance in normal designs in which the second critical speed could be lower than the first would be that of a flywheel mounted on a short shaft. Section II. — Transverse Vibrations — First Critical Speed. A. Neglecting Junctional Resistance. 1. It will simplify the treatment of this question if we first consider the case of a rotor unimpeded by frictional resistances set up by the air and then treat separately the effects produced by friction. The conditions obtaining when a rotor is not perfectly balanced and is rotating are illustrated in fig. 3, where represents the position of the centre line c f the bearings, and C the deflected position of the centre line of the shaft, while Gr shows the position of the centre of gravity of the rotor. thus gives the undeflected position of the shaft centre line and 00 = r the shaft deflexion at any instant, while 0Gr = <? is the error in the centering of the rotor; M<7 is the weight of the rotor acting vertically downwards. The rotation of the rotor about its centre line, i. e. the rotation imparted by the prime mover, is represented by the motion of G around 0, i. e. by the rate of change of 0. and Critical Speeds of Rotors. 129 The whirling of the rotor is represented by the motion of C about the undisturbed position of the shaft centre line, i. e. by the rate of change of a. The " vibr.ition" of the rotor is judged in a general way by the vibration of the bearings as felt when the hand is applied to them. The force on the bearings is that applied along OG by the deflexion r of the shaft, and vibration of the bearings arises through the varying position and magni- tude of OC ; these in turn are due to the motion of the ceritre of gravity G. 2. If the machine is steadily rotating it might at first sioht be thought that OC and CGr would be in the same straight line, so that Gr would be steadily revolving together with G about the undisturbed position of the shaft centre line with an angular velocity n say. At the same time the deflexion 00 = ?' might be changing its value and (neglecting the weight of the rotor) the motion would thus be given by Mr — M?i 2 (V-f e) + ar = 0, or putting crjM = Ci 2 , r + (c x 2 — n 2 ) r = n 2 e, the solution of which is where Nx and y l are constants. We should thus conclude that r would become unlimited Phil. Mag. S. 6. Vol. 44. No. 259. July 1922. K 130 Mr. C. Rodgers on the Vibration in magnitude when the angular velocity n is equal to c x or \f y. , and, further, that the variation in the magnitude of r consists of a free vibration having a periodicity of l /in 2tt V c?-n 2 ' This result for the periodicity of the free vibration would lead to the conclusions that when the machine is not rotating (n = 0) the periodicity is c l5 the same as for the stationary transverse vibration, and that when running at the critical speed (n = ci) the periodicity is zero. There would thus be some intermediate speed where the periodicity of the free vibration corresponds with the running speed, and resonance might take place. This would occur when c 2 — n 2 = n 2 We should thus be led to expect marked vibration when the running speed is — ^ x the critical speed. This conclusion and the argument on which it is based are, however, erroneous. In the first place, the assumption is made that co or 6 is constant and further the condition is omitted that, as all the forces pass through O, the angular velocity about must be constant, or r 2 = h, say. The correct equations for the free vibrations are thus : jV( Cl »-#)r=b,i r 2 = h.) This does not admit of direct solution *, and it is simpler to use rectangular co-ordinates, as we shall now proceed to do. * The solution is, however, well known and is given in books on Dynamics dealing- with Central Forces : — If p is the length of the perpendicular from the centre of force on the tangent to the path, it is known that h 2 dp — —■ — err, p 6 ar giving — = a constant — err 2 , which is the pedal equation of a central ellipse. and Critical Speeds of Rotors. 131 3. With the notation given in fig. 3, it will be seen that th^ position of the rotor is completely defined by the co- ordinates x, y, and 6 {x and y being the co-ordinates of" G) and only three equations are required to express the motion fully ; the value of the whirling angle a follows from the magnitude of the other co-ordinates. The force exerted by the deflected shaft is err, the com- ponents of which are — a{x — e cos 6) along OX and — a(y — e sin 6) along OY. Resolving along OX and OY and taking moments about G, we thus have : M£ + <7(.i'-*cos0) = O, (13) My + (r(y-esm0) + Mg=Q, .... (14) M^+«ra(tfMn0-yco!«0) = O, . . . (15) where k l is the radius of gyration of the rotor about the longitudinal axis through the centre of gravity. In practice the rotor is driven at an average angular velocity co say, which will vary from constancy only by small amounts which we shall find later are negligible. Assuming as a first approximation that the angular velocity is constant = co, so that 6 — cot, (13) and (14) then become, writing cr/M = c 1 2 as before, x + c l 2 x = Ci 2 e cos cot, (16) y + c l 2 y = c] 2 e sin cot — g, .... (17) while (15) becomes an identity. The solutions are ■jjrnN^in M-yi)-f — ^-^coscot, . . . (18) y = N 2 Cos ('•!* — 71J+ 2 Cl e 2 sin cot-g/c^, . (19) C'i — CO where N 1; N 2 , and y l are constants. 4. The above results give the motion of the centre of gravity G. The motion of the centre of the shaft C is to be found in a similar way, still on the assumption that the K 2 132 Mr. 0. Rodgers on the Vibration angular velocity of rotation is constant. If x 1 y' are the co- ordinates of C, ,v = x' 4- e cos o>£, giving the equations ■// = t/ -\-e sin cut it +Ci'V =W^COSG)^ (20) y' +cfy' = oo~e sin t»£ — <?, .... (21) the solution of which is : 9 x' = TX l sm{C l t-y 1 ) + -p^GQ8<ot, . . . (22) v' = N 2 cos (CV — 7i) h 2 — -^sin w^— ^/c*! 2 . . (23) ('l — CO 5. These equations are the same as for a perfectly balanced rotor with a weight attached to it of such small value as not to affect the position of the centre of gravity, or the down- ward pull due to gravity, but producing a force of Mco 2 e. In other words, we can treat the unbalanced rotor as if it were a perfectly balanced rotor with a force Mw 2 <? attached to it, and as this mode of presentation is easier to follow than the former, we shall employ it in the remainder of the paper. 6. It will be seen that the solutions for the motions of the centre of gravity and the centre line of the shaft are the same, except that the former has an amplitude - 2 — 2 , and CO" € the latter an amplitude of — ^ - 2 , so that they differ by the amount e ; this shows that 00 and CG are in the same straight line when not running at the critical speed. 7. The solutions show firstly that the motion takes place about the position Ma q which we have seen is the statically deflected position of the centre of gravity of a perfectly balanced machine. It is sometimes contended that as the speed increases the rotor shaft tends to straighten out, but there is no indication in the present treatment that this is the case. 8. The free vibration x — N t sin {i\t — 71), y = N 2 cos fa*- 7O, is the same as for a perfectly balanced rotor and has a and Critical Speeds of Rotors. 133 frequency the same as the frequency for stationary vibra- tions, and, as we shall presently see, ii' expressed in R.P.M., is the same as the first critical speed. It is, therefore, inde- pendent of the speed of rotation of the rotor, and there is no possibility of resonance occurring at -y^ X the critical speed as suggested by the erroneous method mentioned earlier in this section. The free vibration itself is thus represented by two components having the same frequency, but different amplitudes; it is therefore a central ellipse, the centre being at the point y= . It will be shown later that the free vibration is damped out by friction, so that it has no importance in practice. 9. The forced vibration for the centre C of the shaft is given by x= —. — . 2 cos cot, (24) Ci — G) we £=_-—_ sin cot-a/cj 2 . . . . (25) 6*i — ft) This naturally has the frequency corresponding to the angular velocity of rotation o>, and has a maximum value of we c , 7. tor each axis. Cj — CO' The motion of the centre is thus a whirl, the radius or amplitude of which is proportional to the out-of-balance and is zero when the rotor is perfectly balanced. A perfectly balanced rotor, therefore, cannot whirl in the manner ex- pressed by equations (24) and (25). 1 The amplitude of the whirl is also proportional to -j 2 and it thus becomes a maximum when * (i)= +Ci. (The sign + merely indicates that the rotation may be in either direction.) This value of co gives the first critical speed, which is thus the same as the stationary frequency for transverse vibra- tions. Reasons will be given later why the radius of whirl does not become infinite at the critical speed, i. e., why the shaft does not break when the rotor reaches this speed. 134 Mr. C. Rodgers on the Vibration 10. At the critical speed where co 2 = c\ 2 equations (20) and (21) become : x + c 2 x = c'i 2 e cos C]t, y + cfy = c 2 e sin erf — g, which admit o£ a solution not involving infinite values, namely, # = £ — sin c x t = *f «*(«.«- |) (26) y=—i^ r cosc i t — (?/c l 2 <\e . / 7r = iysini^-- )-#i 2 (27) Equations (26) and (27) thus give the motion at the first critical speed when friction is ignored. They show that the component along each axis has an amplitude which con- tinually increases in proportion to the duration of the motion, in other words the motion at the critical speed is a spiral of continually increasing radius. 11. From (24) and (25) it will be seen that the phase difference of the motion with respect to cot changes from zero to 180° as co passes through the value c l} i. <?., as the speed passes through the critical ; (26) and (27) show that at that speed the motion lags behind cot by 90°. It will be seen later that when friction is taken into account, the lag- increases gradually as the speed increases, being still 90° when &> = <?!. 12. Up to this point we have treated the angular velocity 6 as a constant = &), as on the average it will be in practice. Suppose now that it varies slightly from constancy, so that the angular position cat becomes cot + u, where u is so small that its square can be neglected, and we may write smu = u and cos?/ = l. We then have cos 6 = coscot — u sin cot and sin = sin cot+u cos cot, also = u. Substituting these values in equations (20), (21), and (15), and writing -^ =c{ 2 as before, we get x -f Ci 2 x = w 2 e cos cot— co 2 en sin cot, . . . (28) y + c l 2 =y co 2 e sin cot + co 2 eu cos cot— g, , . (29) MM + cre(,T sin cot —y cos cot + ant cos cot + yu sin cot) = 0. (30) and Critical Speeds of Rotors. 135 As a first approximation we substitute in (30) the values already found tor .t* and ?/, as given in (24) and (25), for the forced vibration and thus obtain (neglecting u in comparison with 1 radian) : 2 M/iyzV + o-tf 2 — 2 2 a = —<reg]c 2 cos cot, . . (31) giving for the forced vibration (the free vibration may be ignored as it will be damped out) : e(c x 2 -co 2 ) ,_ OA Z ' = * 2 2 27 2/ 2 2\-gCOSO)t. . . (.32) At the critical speed co = c^ and ^ = 0, z. 0., the angular velocity of rotation is constant ; at other speeds than those in the neighbourhood of the critical, the term co 2 e 2 c 2 mav De neglected, as e 2 is much smaller than kx 2 , so that u = — 7 -„ . - 9 co§ cot (33) faf CO 2 The variation in the angular velocity has thus the same frequency as that of the rotation itself, but is very small in magnitude, as will be seen from the following figures. For a turbo-generator rotor balanced to about 1 oz. at radius A'i per ton weight of rotor, e\k x will be about 3 x 10 ~ 5 , and for a machine to run at 3000 R.P.M., £ x will be about 50 cm., and co is, say, 2-7TX 50, 3xl0~ 5 981 U= ^r — x r—s — ^ — ^ cos cot oO 477-- x 50 x 50 = 6xl0" 9 coso)£, that is the vibration is very small, the amplitude being only 6 X 10" 9 of one radian. It should be noted that u is proportional to e and to g ; this variation therefore arises through the action of gravity on the rotor when not perfectly balanced, and the variation will be absent if the balance is perfect. Substituting the 'value of a in (28) and (29) in order to find the effect o£ the irregularity on the displacement : — e 2 & + c 2 x = co 2 e cos cot + 7 2 g sin cot cos cot, e 2 y -f c{ 2 1/ = co 2 e sin cot— — q cos 2 cot — g, 136 Mr. C. Hodgers on the Vibration or 1 e 2 x + c?x — cw 2 e cos oot-i --=— g sin 2g)£, V + Ci 2 y = a> 2 e sin cot- - -^ g cos 2o>*-^ 1 + -^j . The solution is (for the forced vibration) co 2 e cos cot 1 e 2 1 . _ , rt JN I= l?^ + 2V^i' SmH (34) &> 2 £ sin &>£ 1 e 2 1 _ g /^ 1 e 2 \ , „ N The irregularity has thus two effects on the main whirl : firstly, the static deflexion is increased by a small amount, and secondly, there is superimposed on the main whirl a ripple o£ double frequency, which rises to a maximum at half the critical speed. But the effect is very small, and may not be noticeable ; in any case, as the double frequency effect depends on e it cannot appear when the machine is well balanced. There is, however, the possibility that a rotor which is not perfectly balanced may show vibration at half the critical speed due to the action of gravity, although gravity would produce no such effect at the full critical speed. A vertical spindle rotor is not of course subject to the action of gravity in this sense, and if it vibrated specially at half the critical speed, the cause must be sought elsewhere. 13. We shall now consider some further possible causes of subsidiary critical speeds or speeds where marked vibration may appear other than the normal calculated critical speed. 14. An important case is that of a rotor slotted for a 2-pole winding or with a shaft in which a key-way is cut, where the rigidity of the rotor is greater in one direction than in a direction 90° away, so that if the shaft is rotating, the stiff- ness in the direction of any one of the axes is not a constant er but a 4- e cos 2cot, where e is small in comparison with cr. We shall assume that the rotor is perfectly balanced (e = 0) and to simplify the examination shall first consider the vertical motion only. The equation is : M.y + cry=— M^— yecos2a>t. The first approximation is : y=— — = -#i- id Critical Speeds of Rotors. 137 Inserting this on the right-hand side of the equation and sol vino- we get y=— ' il-f - , M „gos 2 cot y ,y a [_ cr — 4M&) 2 J = -^'i 2 { 1 + €/'M cos 2w/ (36) Cl --4:C0 2 J' ' There is thus a double frequency vibration about the statically flexed position, which has a maximum when (0= 2 c u that is, at half the critical speed. It is evident that such a motion must have a tendency to arise if a rotor is unsymmetrical as regards its rigidity, for in such a case when the shaft rotates the deflexion will be a maximum or minimum twice every revolution, and if the frequency of the consequent up and down motion is equal to the critical speed there will be resonance ; this will be the case whether the rotor is perfectly balanced or not. It is thus possible for a perfectly balanced rotor which would be quite steady at the critical speed to show marked vibration at half that speed. If the normal running speed is above the critical the forces called into play at half the critical speed will be very small and may give no appreciable effect, but if the running speed is in the neighbourhood of half the critical speed vibration might arise. Fig. 4. u ,--Ti£t 15. It is worth while to examine the motion a little more fully as there will evidently be some vibration in the hori- zontal plane also. Let C (fig. 4) be the position of the centre line of the shaft and OA, OB two axes at right angles rotating about with the same angular velocity co as rotor. Let the co-ordinates of C be a and b with respect to OA and OB and u and v the corresponding velocities along those axes. 138 Mr. C. Rodgers on the Vibration Then u =« — &©, and the accelerations are along A : u — vo) = a — 2/;<y — aw 2 , along OB : v + uco = b + 2aco — bay 2 . If the force required to produce unit deflexion in the shaft is <r + e along OA and a — e along OB, and we resolve along OA. and OB, the equations are : M(« — 2bco — aco 2 ) 4- (<r + e)a = — M<? sin g>£, M("i + 2aco — bco 2 ) + (o— e)6= — 1% cos cot ; that is, {M(D 2 -ft) 2 ) + (o- + e)}a-2Mr«)D/>=-M^sinft)^ {M(B 2 - co 2 ) + (a -6)}b + 2McoI)a=-Mc/ cos cot, giving [{M(D 2 -a) 2 ) + cr} 2 -6 2 + 4MVD 2 ] a = — M.g(a — e— 4&> 2 ) sin o>£, [{M(D 2 -o) 2 ) + cr} 2 -€ 2 + 4M 2 a) 2 D 2 ] 6 == — M#(y + e— 4« 2 ) cos cot, The solution is (neglecting e 2 in comparison with cr) This gives the position with respect to the rotating axes ; the position with respect to the fixed axes is that is x = a cos cot — b sin cot, y = b cos cot -{-a sin cot, This result is the same for y as obtained in (34) by the and Critical Speeds oj Rotoi-s. 139 method of approximations ; it shows that there is a similar motion of equal magnitude and 90° out of phase along the horizontal axis, so that the motion is a circular whirl of double frequency, which rises to a maximum at half the critical speed. 16. When the rotor is out of balance the equation for the vertical motion is Mi/-f cn/ = M&> 2 £sin cot — Mg— ye cos 2cot. The first approximation is given by (25) and inserting this on the right-hand side of the above equation we get : 1YT M y + crfj = Mmh sin cot — Mg + — e cos 2cot . e Mco 2 e , . . ' , / qo n + 5" Tv/r «{sin m* --sin .•5ft>£|. \^) ! (7 — Ivico The first three terms on the right correspond to the main whirl and the double frequency whirl already dealt with. The last term on the right will give in the solution a triple frequency vibration, viz. : 1 Mor<? e . _ n M 2 nil/I 2 sin 6wt i I ff-Mr a — 9 Mar which has a maximum value at ^ the critical speed. This vibration is, however, proportional to e, the out-of-balance force, and cannot arise in a perfectly balanced machine. The remarks made as to the limited conditions under which the double frequency vibration might arise apply with even greater force to the triple frequencv vibration as the damping- effect of friction will be correspondingly greater. 17. Another case of interest is that in which covers or sleeves are mounted on the rotor, or the rotor has slots in the periphery for an exciting winding, closed by pressed- in keys ; the closeness of these force tits will vary with the deflexion, and the deflexion of the shaft may therefore be not quite proportional to the force applied, i. e., the force to produce a deflexion % will not be o\r, but say a(x + ex 3 ), where e is small in comparison with unity. (The expression for the force must contain odd powers of x only as the rigidity is symmetrical, that is, the same numerically for the same numerical value of x whether x is positive or nega- tive ; if even powers were included this could not be the case as an even power of x is always positive even if ,c itself is negative.) 140 Mr. C. Rodgers on the Vibration The equations then are, putting the small quantities on the right-hand side : x + c 2 x = (o' 2 e cos cot — c^ex?, y ~t" G iy — ( ° 2e *> m Mi —9 ~~~ c i 2 ^/- Neglecting the small quantities, the forced vibration is as before given by we X=-^-—: 2 COS(Dt,. ..... (40) 9 co e y=J*t^p ainht-g/d*. . . . . (41) Inserting these values on the right-hand side o£ the original equation, we get after some reduction : — x + c 2 x = {co 2 e - %<'i 2 ep 3 } cos cot — ^cfep 3 cos Scot, ij + Cl 2 y= -g + cfey^pP+yJ) + {co 2 e + c l 2 ep{'2i/ 2 t fp 2 ) } sin cot — Ci 2 €p 2 g cos 2cot - r jc^ep 3 sin Scot, where p= — J = c/ — OT and i /o =gjc l 2 . Solving these equations we get : * = /> ll-jc^e ^^ J cos tot-\c v 2 €-^_ — 2 cos 3u*, Ci 2 € 2 • p 2 yo cos % w t C 2 -± * c i 2 e + i • ~9 — rr~i P z sin 'dot. C] 2 - yco 2 r Examining these terms in turn we find that the centre of motion is now at the point #=0, instead of the point # = 0, y=- - y . This indicates that the centre of motion rises, i. e., the shaft straightens out slightly, as the vibration increases. and Critical Speeds of Rotors. 141 The main vibration, represented by cos cot and sin cot, shows a slight change in amplitude, but as before the maximum occurs at the critical speed. The term in cos 2cot indicates there is a double fre- quency ripple in the vertical motion y (but not in the c 2 e horizontal component x) having an amplitude 2 2 /° 2 ffo? that is, ' l . ,.( ., .,) Vn- This rises to a maximum at half the first critical speed (when the amplitude changes sign) and again at the first critical speed (when the ampli- tude does not change sign). Noticeable vibration may thus occur at half the critical speed, but it will take place principally in the vertical plane. Both components show a triple frequency vibration ex- P'essed by cos Zcot and sin Scot, which reaches a maximum at one-third the critical speed, and the amplitude of the vibration changes sign at that point. This vibration also has a maximum value at the critical speed. Points of marked vibration due to lack of proportionality in the deflexion can thus only show themselves when the machine is out-of-balance, and if they become appreciable at all will only occur at half or one-third, etc., of the critical speed. If, however, these fractions of the critical speed correspond to low running speeds, the forces may be so small as not to produce any noticeable effect. 18. It thus appears that subsidiary critical speeds are only to be expected at half or possibly one-third of the calculated first critical speed, and only then when the subsidiary critical speed is high enough to make the forces appreciable — for example, in the case of a turbo alternator when the speed indicated by the calculation approaches the running speed. All these effects should disappear with perfect balancing, excepting that due to lack of uniformity in the resistance of the shaft or rotor to bending in directions perpendicular to its axis, such as might arise through two-pole slotting of the rotor or through a key-way in the shaft. The forces tending to produce vibration are small, and the vibrations arise through a kind of resonance ; as there is a good deal of damping due to air friction and to the move- ment of the shaft in the bearing where the oil exercises a strong damping action, the vibrations may not arise at all. This question is gone into more fully in a later section. 19. We have now to consider some cases where resonance may arise from causes outside the machine itself, and two 142 Mr. 0. Rodgers on the Vibration classes may be noted, firstly where there is an irregularity in the torque applied to the shaft, and secondly where there is resonance with masses outside the machine. 20. Irregularities in the torque driving the machine may arise, for example, through variation in the steam admission or through a fluctuating electrical load. The result of fluctuation in the torque will be a corre- sponding fluctuation in the angular velocity of rotation so that the angular position, instead of being cot, will be cot-\- € sin pt, where e is a small angle and p is an angular velocity corresponding to the frequency of the disturbance. Then cos {e sin pt) =1 and sin (e sin pt) = e sin pt. The equa- tions then become, taking the small quantities on to the right-hand side : x -j- c 2 x = co 2 e (cos cot — e sin pt sin cot) , jj -f c-iy = (o 2 e (sin cot 4- e sin pt cos cot) — g ; that is, x + c 2 x = &) 2 <?{cos cot + e/2 (cos o> +■ p . t — cos co — p . t) }, . (42) y -\- c^ 2 y — co 2 e{sin cot + e/'2{>in co-rp . t-\-s\\\co — p . i) }. . (43) The main vibration is the same as before, but there are two small vibrations superimposed ; the one has a frequency corresponding to co-\-p and a maximum when <w = 6 , 1 — p, the other a frequency corresponding to co— p and a maximum when a>=c 1 + p. This shows that the vibration may have a maximum at speeds corresponding to the sum of and to the difference between the critical speed and the speed corresponding to the frequency of the disturbing fluctuation. So that if dis- turbed running show itself at such a speed that it cannot be otherwise explained, a cause may be sought for in this direction. 21. The other variety of resonance mentioned is that where, for example, the foundations are not sufficiently rigid and the machine as a whole is vibrating so that there is resonance between the rotor on its shaft and the machine on its foundations. A similar case would be that of a machine rotating in or near a building which itself shows marked vibration, possibly in certain parts only, corresponding to the vibration of the machine. Both these cases are similar in principle and may be illustrated by supposing the whole machine to be mounted on foundations having some elas- ticity. If then M x is the mass of the rotor and M 2 the and Critical Speeds of Motors. 143 effective mass o£ the machine and that part of the founda- tions which moves with it, and the forces required to give unity deflexion are in the two cases cr, and a 2 respectively, the equations of motion are as follows : Mii/! + o-, (y, - y 2 ) == M 1& rV sin pt, M 2 i/ 2 + 0-22/2 - o"i (yi ~ Vi) = 0. giving for the amplitude of the forced vibration • • (44) where C\ 2 = a M,' u> 2 e(cx 2 m + 2 '2 ~ CO 2 ) CO 4 - ■ (o 2 {Ci 2 m-\-c 2 + «I 2 J + CiV 9 °2 ~M 2 ' in = M 2 • Points of marked vibration may thus occur at either of two frequencies given by putting the denominator = ; M these frequencies will therefore depend on the ratio ~ as well as on C\ and c 2 , and may thus have almost any values. For example, if c 2 = i\ and ?n = 0'2, i.e., the mass of the machine and foundations is five times the mass of the rotor — fi> s g(l-2c 1 2 ~o> 2 ) • y ~a> 4 -co 2 (• 2 2 x2•2 + r I 4, and a maximum occurs when ta=c 1 xl'25 or c 1 x 0*80, that is, at speeds 25 per cent, above and 20 per cent, below the calculated critical speed. If M 2 , the mass of the machine and foundations, is very large in comparison with M 1? the mass of the rotor, the denominator is very nearly equal to (co —c{) X (a> — c 2 ), which shows that in such a case the two speeds where marked vibration may occur nearly correspond to the natural fre- quency of the rotor and of the machine and foundations respectively. But as the numerator is also small the vibra- tion might not appear if considerable friction is present. It: vibration should occur when ft) = c ] /v / 2, which is, as mentioned above, sometimes thought to be a critical speed, this might indicate that there was resonance with the founda- tions or some structure outside the machine, in which case, 144 Mr. C. Rodgers on the Vibration putting co 2 = c 1 2 /2, c 1 2 -2 W{l + m) + r 2 2 } -f4c 2 2 = 0, that is c 2 = \ ' — g — c„ so that if, for example, M 2 were large in comparison with Mi and m is therefore small, c 2 = c 1 /V2 = c l xO-707, but if, as in the former example, M 2 = 5M l5 c 2 = CjL x 0-836. B. Transverse Vibrations with Friction. 1. The frictional resistance opposing the motion of the rotor may be considered to consist of two parts. The first part opposes the rotation of the rotor about the centre line, C, of the shaft, and this is counteracted by the torque supplied by the turbine, or, if the turbine is cut off from the steam supply, it tends to bring the set to rest ; it has no retarding effect on the whirling. The second part opposes the whirling only, and it is with this that we have to deal. 2. It is not known how the frictional resistances opposing the whirling vary with the speed, but it seems likely that they vary with the square of the speed at least. We shall, however, first consider the case where the resistance is assumed proportional to the first power of the speed, as the motion is then simpler to work put, and there is an inter- esting electrical analogy, which enables the motion to be more readily followed. The resistance to whirling is in opposition to the path of ds the rotor centre, so that if -r- is the speed of the centre in any direction, the frictional resistance is M//,f , ) , and the (ds\ n dx 7 - ) t- and f(xs\'" av M-fjulj.) -/-, that is, Mfzs^x and M/^" 1 ?/, where fi is a constant. 3. In the particular case we are about to consider, n — 1, and the components are therefore M/xx and Mfiy. and Critical Speeds of Rotors. 145 The equations of motion are therefore Mai -I- M/Xci; + ax = Ma) 2 e cos tot, M.y + Mfiy + ay = Mco 2 e sin atf — M#. Electrical engineers will notice the similarity between these equations and Jj'q -f ~Rq + p- g = E sin o)£ + E 0j which holds for a circuit comprising an inductance L, a resistance JEfc, a capacity K, an alternating E.M.F. of maximum value E, and a steady E.M.F. E , q being the charge in the condenser at any time. Thus, mass is equiva- lent to inductance, capacity to deflexion per unit force, and applied E.M.F. to applied mechanical force. The solutions of these equations are, as is well-known : — q = ~$e~ t > T sin (pt — <f>) E • / , ■ i E \ H r /-. it- t — ro — TV^r sl n ( cot — tan zmy 7 — V{(l/Ka)-La)) 2 +li 2 } V 1/Kft) — La>/ + KE , (45) ^ = Ne _i/T sin (pt — <f>) Mco 2 e . / ' , _, Maw \ + ^{(,-M.f+MV^} sm r " tan " ^Mo?j -M<?/<r, (46) that is, r/ = Xe~ f/T sin pt — <j> + /7-5 — -072 , » 2 sin ( wt -tan" 1 2 A6> 2 ) — ^/q 2 , , • • • ( 47 ) where 2L 2M ^ = Vl/LK-1/T 2 or W/M-l/T 2 , There is therefore in both cases a free vibration having a frequency slightly less than the natural frequency of the system, but independent of the frequency of the applied E.M.F. or of the speed of the rotor. This vibration is damped out by friction. Phil. Mag. S. 6. Vol. 44. No. 259. July 1922, L 146 Mr. C. Rodgers on the Vibration © The forced vibration is permanent and has a value which is proportional to the applied E.M.F. or out-of -balance force. The vibration is a maximum when co = c u that is the critical speed is the same as the stationary critical, as in the case where friction is ignored ; the amplitude of the vibration at .Ee.. the critical speed is p or — , that is, is equal to the applied E.M.F. or out-of-balance force and inversely proportional to the resistance. The frequency of the forced vibration is the same as the frequency of the E.M.F. or of the rotation, and the charge or displacement lags behind the E.M.F. by an amount depending on the frequency or speed and on the capacity and inductance or elasticity of the shaft and mass of the rotor. The lag is zero when the frequency is low, but increases to 90° at the critical speed, which, as will be seen, is that corresponding to the natural frequency of the system, while at very high speeds the lag increases to 180°, in other words the force is in opposition to the displacement. The change is similar to that occurring when there is no friction except that in the present case the change is gradual instead of taking place suddenly at the critical speed. It will be noted that in both cases the vibration takes place about the statically deflected position as a centre. It is evident in both cases that a large static deflexion would increase the tendency to break down, in the one case by puncture or flashing over of the condenser, and in the other by fracture of the shaft. 4. If the resistance to whirling is proportional to the square of the speed, that is n = 2, the equations are : x + fi'sx 4- CiX — co 2 e cos cot, y + fisy + c 2 y = co' 2 e sin cot — g. The free vibration (i. e. the vibration when g = or the rotor is perfectly balanced) cannot be expressed in simple terms, but as it will be damped out, as before, it is not of interest. The forced vibration (i. e. the vibration due to the rotor being out-of-balance) is given by .£ = R cos (&>£ — <£), (48) y = R sin (cot - (/>) — g/ Cl 2 , .... (49) and Critical Speeds of Rotors. 147 where R is the radius of whirl of the value : co e and tan<f> = \ 9 (51) r Ci— co z v y The lag of the displacement behind the force is in this case proportional to the actual deflexion, and in this respect differs from the result obtained in (47). This is of interest as it shows that since the radius of whirling is for a given speed dependent on the out-of-balance, the phase lag will be smaller the more perfectly the machine is balanced ; in the former case where the friction varied as the first power of the speed, the lag was independent of the amount of out-of-balance. The maximum deflexion occurs, as before, when &> = c l5 that is, when the speed is equal to the stationary critical speed. 5. It is impossible to draw any conclusion from these formulae as to the real angular advance corresponding to a given speed, as it is not known how the frictional resistance varies with the speed. We can, however, say that if the machine is rotated first in one direction and then in the other, the position corresponding to the out-of-balance will be mid-way between the points of maximum deflexion. When balancing a machine in the running condition it is usual to hold a pencil or chalk against the shaft so that a mark is made on the shaft at a point corresponding to the maximum deflexion. If there were no friction and the speed were not the critical speed, this mark would be in phase with the heavy side of the rotor below the critical speed, and 180° oat of phase with it if above the critical speed. But it will be seen from (51) above that the actual position of the mark depends both on the amount of friction and on the amount of out-of-balance. At the critical speed the heavy side of the rotor should be 90° out of phase with the mark on the shaft, but the actual position will be uncertain, as the angle varies rapidly with departure from the critical speed, and it is not usually possible to judge exactly when the machine is running at the critical speed. 6. There is another reason why the position of the mark on the shaft is somewhat uncertain. Referring to fig. 3, if we ignore all other vibrations than that corresponding to the variation in <j), we get by taking moments about G : MkM+crre sin 4> = (52) L2 148 Mr. C. Rodgers on -the Vibration If <j> is a small angle this becomes : M/q 2 <£ + am/)=0, the solution of which indicates a periodic motion having a time o£ vibration of T = 2tt^ Mk, 2 = 2 *V§e < 53 > In an actual machine for 3000 R.P.M. we shall have figures of the order of: — 7^ = 50 cm., k A \e — 3 x 10 4 , c 1 = 27rx30, say, while r may be of the order of 1 mm., so that T __ 2it /50 x 3 x 10 2vrx30 V 0-1 = l'4Xl0 2 sec. or about 2 mins. As the time of vibration is very long in comparison with that of the other vibrations occurring, it will be almost unaffected by the latter, and the assumption that the other vibrations can be ignored, which was made in deducing (52), is therefore justified. For larger values of e corresponding to less perfect balance and for deflexions of greater magnitude, T will be corre- spondingly less, and will be greater the more perfect the balance. If friction is ignored, r becomes infinitely great at the critical speed and T becomes zero, and although this can never be the case in practice, it is clear that T may have a value of two mins. or more down to something considerably smaller. In other words, if the rotor is disturbed from its position of equilibrium by any chance external cause, it may take a considerable time to settle down, and during that period the position of the mark on the shaft will vary considerably from its normal position. 7. At the critical speed the lag is 90°, and the vibration is also a maximum, but the sharpness of this maximum will, as indicated above, depend on the frictional resistance to whirling. In addition to this it will also be influenced by the condition and design of the bearings, as the oil in tha bearings exercises a considerable damping influence and aL-o introduces a further complication as follows : When the speed is low and the vibration therefore small and Critical Speeds o) Rotors. 149 in magnitude, the film of oil in the bearings will allow the shaft a certain amount of play ; this will increase the effec- tive length of the shaft and lower the critical speed; with increasing speed the vibration will therefore start up fairly smartly. As, however, the speed increases and the vibration becomes greater, the shaft may bed hard up against the bearing bush, and increased deflexion will decrease the effec- tive length of the shaft, and so raise the critical speed. As tbe speed is further increased a similar state of things is gone through, so that at a certain point the vibration will die down more quickly than if there had been no film of oil. The effect of the oil in the bearings is thus to give an added amount of friction to whirling, and at the same time flatten the maximum peak of the vibration, that is, the vibration will start up and cease fairly smartly and remain more or less constant throughout a fair range of speed. If, however, the film of oil is sufficiently thick or the balance sufficiently good, the vibration may not show itself at all, although it might do so with the same out-of-balance, if the film of oil were thinner. Section III. — Oscillatory Vibrations — Second Critical Speed. 1. Oscillatory vibration may arise in two ways, either through lack of balance or through vibration transferred from the transverse motion. 2. The lack of balance referred to is of the skew type, that is, is equivalent to a pair of weights at opposite ends of the machine, and 180° apart, giving an out-of-balance couple when the machine rotates ; such an out-of-balance will not show itself when the machine is being statically balanced on knife edges, and can only be corrected through observations when the machine is running. 3. Vibration can be transferred from the transverse motion only when the machine is unsymmetrical in the sense that a force applied to the centre of gravity at right angles to the shaft gives a displacement which is not parallel to the centre line, that is, in those cases where, on the static deflexion diagram, the shaft in the deflected position is not parallel to the centre line of the bearings. 4. The form of out-of-balance mentioned will produce a couple rotating with the machine, that is, a couple alter- nating with the frequency corresponding to the running speed. Vibration transferred from the transverse motion ma}-, however, be of the frequency corresponding to the 150 Mr. C. Rodgers on the Vibration speed of the machine, but may also be of double frequency arising through any of the causes we have discussed. Further, when the normal frequency oscillation has estab- lished itself, a double frequency oscillation may start up owing to bipolar asymmetry or some of the other causes mentioned in connexion with the transverse vibration. It is therefore necessary to consider in the oscillatory motion forces both of the actual frequency of rotation and of double frequency. 5. We have found that the transverse motion can, with sufficient accuracy, be considered the same as for a perfectly balanced machine with an out-of -balance force attached to it. In the same way we shall treat the oscillatory motion as being due to an out-of-balance couple of the frequency corre- sponding to that of rotation or a multiple of that frequency acting on an otherwise perfectly balanced machine. 6. In the diagram fig. 5 let G be the centre of gravity of the rotor and GL the direction of the centre line of the rotor twisted from its normal position by an angle f = L(}Z, where GZ is the direction of the centre line when not vibrating. The direction cosines of the centre line GL with the axes GX, GY, and GZ are respectively f , 77, and J ; if L is a point at unit distance along the shaft from the centre G, then J, 7), and f are also the co-ordinates of the point L, as shown on the diagram. If the moments of inertia of the rotor about the shaft centre line and about the line at right angles to it, through and Critical Speeds of Rotors. 151 the centre of gravity, are A and B respectively, the angular momenta are *: about GX, ^B^-SHAwf, about GY, ft 2 =B(?f-ff) + A©?7, about GZ, A 3 = B(^-^) + Ao)f. In the actual case the angle yjr is very small, so that we can put sin-v/r = ^ cosi/r=l, and f=l, f=0j also the products f 77 and 77^ are both negligibly small. We thus get : 7*i =— B?7 + A&)f, 7l 2 =:B| + Aft)>7, 7i 3 = Ao). * Another and perhaps more legitimate way of deducing these equations is as follows : — If a lt wo, and o) 3 are the instantaneous angular velocities about moving axes 3rX', GY', and GL fixed in the rotor and moving with it : consider the iistant when GX' is perpendicular to GL and GZ (cf. fig. 5), and let )e the angle between the planes LGZ and YGZ. Th«nw l =— -^ w 2 = 0sin^, and w, = w. Th( angular momenta about GX', GY', and GZ are Bo>i, Bw 2 , and Aw. The anuular momenta about the fixed axes GX, GY, and GZ aie : Ai = B{ w l cos 0+w 2 cos \p sin 0f4 Aw 3 shuf/sin 9, h 2 = B { — o>i. sin 9 4- w 2 cos \p cos 9 \ -j- Aw 3 sin ^ cos 0, 7i n — B{ — w 2 sin^}+Aw 3 cos^ ; that i; hi= B{— ^ cos 9-\-9 sin ip cos 4> sin 0} + Aw 3 sin;// sin 9, h 2 = B{ -4/ sin 0+6/sin^cos4/cos0} + Aw 3 sin^cos0, h = — B0 sin 2 iff+Awj cos 1//. Al.o S = sin^sin0 and £=ij/cosi//sin 0+0 sin ^cos 9, J7 = sin'4/cos0 and r/ = ^ cost// cos 0—0 sin ^ sin 9, £ = cosip. and £== — i£sin^; so thit tlK—Ki = —yf cos 0+0 sin cos^sin0. Zi-ZZ— i\j sin 04-6* sin ^ cos $ cos 0. % n — ■,)'%= — O&m 2 ^. ty substituting these values in the equations for A t , A 2 , and A 3 , we obain the relations given above. 152 Mr. 0. Rod o'er s on the Vibration (54) (55) (56) The force o£ restitution of the shaft is : about GX = ^, about G-Y= -—«£", about GZ=0. If the out-of-balanee couple is ra 2 , the moment of the couple is : about GX = — ra 2 sin pat, about GtY = to) 2 cos pat, about GrZ —nil, where p = l or 2. The equations of motion are thus : — B rj — fcr) + Ao>f = — ra 2 sin pat, Bf+/cf + Aft)^ = TCO 2 COS Jt?&)£, . . Aai=0; or putting D for oj/d£, (D 2 + c 2 2 )rj — maD^= rco 2 /B . sin pat, (D 2 -}-e 2 2 )f -f mo)D^ = T&) 2 /B .cospw^ Ai = 0, where c 2 2 =^ and m=~-. The last equation A&> = gives -co = const., that is, trBre is no fluctuation in the angular velocity of rotation. The other equations give {(D 2 + c 2 ) 2 + m 2 a> 2 D 2 } f = ra 2 /B {{c 2 2 -p 2 a 2 )-mpa 2 } co\pat, and a similar equation for f in terms of sin pa)£. , 7. The free vibration is not of importance, as it wil be damped out as before, but it is of interest to examira its value as illustrating the effect of gyrostatic action oil the motion. The free vibration is of the form x = Ni sin (qt — (f>i) ; y — N 2 cos {qt — cf) 2 ) , where N 1? <f>i, N 2 , cf> 2 are constants and + q — ±{ma+ \/m 2 a 2 -\- 4c 2 2 }. '7) and Critical Speeds of Rotors. 153 There are thus two natural frequencies of whirling, depending on the direction in which whirling takes place. For example, if the machine is running at what we shall presently see is the critical speed, namely, mo) = c 2 — \/ K , there are two possible frequencies of whirling for the free vibration, viz. : g=l*618 C 2} or 0*618 C 2 . 8. The forced vibration is given by * ™V B » (J ,B- A) -Btf. . «****'•' * • (57) poa with a similar equation for r; in terms of sinpcot. The amplitude of the vibration is a maximum when po> 2 (pB-A)-Bc 2 2 = 0, or E) 2 fe 2 = / V A^ = I 1 M • < ' ( 58 ) ' 2 p(pB — A) p(p-m)' 9. This enables the second critical speed to be calculated without difficulty, and fig. 6 gives the necessary curves for reading off the proportional values directly. The ordinates give the values of co/c 2 and the abscissae the values of the ratio A/B. The method of using the curves is as follows : — 1. Work out the radius of gyration ki about the shaft centre line. 2. Work out the radius of gyration k 2 about a line perpen- dicular to this through the centre of gravity. 3. Work out the ratio of A/B or k 1 /k 2 ; for turbo-generator rotors its value is usually between *2 and *4 and for flywheels up to 2'0. (This gives the working point on the horizontal axis of the curve.) 4. Work out the *' first " critical speed in the usual way and multiply by ljk 2 , so as to obtain the stationary "second " critical (where Z = half the distance between bearing centres). •5. To obtain the second critical speeds read off from the curves the figures given on the vertical axis and multiply by the stationary critical found from (4). 154 Mr. C. Rodgers on the Vibration 10. The couple causing oscillatory motion will usually be of a rotating type having components represented by the Fisr. 6. 2-5 rax .': 1". ■'.:!. • ■ 1 : > ', > ' i .. ...,_,....!. i : ,.j_.I .;...,... |._i..:. . . ,. ..._.^.L_ 1 : • • i ' i ' -_.Li_!:".U_.i 1 .: -.-.J : . : i/-i It i: - . r r ■ 1 1 !i_] ; J j :' . | i. " 1 . ' l" ; l~~ : . !' "} .-: }-:[: ■". _ : ** _______ .. ...i . . ..;... . ,. . /..[. ...]„L_ V't-r- r/ r " : i t "j •T-™p j ■ ' - S.-_o^_7-.7'\.7 Sr-jx«£^rT:~- ' 7 ;'7"" '" j , ! : ... ;.. . j ,'. 1 ... . hj< — l^te - .: ; . •: / ■ ■ h(A—~ i -j^ttfjf- --!■ ■ ; : i J, - : - ■ *°|n'-'2"T''~ • J • pr ; _ ...... . _| y.jL.^...- . . /' ; "". i ' " ! ..:_... ;.. _• .. . .. i . : ,• i'2 ; ! ; TP'l • : H ho _ [ ..'.'• ^ PPj- '9 7.7- )_T~ *? 1 .... ,. . : J . . j ..s.j ■i 7|7„ ;„j-... T77] — ... . :. . . . ... • . .|_al_J_!. "TT ~"~~ |~r~~ * •-' ; . tijttc '•' • ■ ! hi ! : ! ; ! 6 l 1 ! : Li L ; -:— | ■■--•- ;• ']'— ? ■ ■+•• ; ' f. - jftj- 1 7 4 B :. /' I >:. .,.:/ T"_> ""V""^. -5 •_ 7 V""-?, /•X"'^ A3 ~* >.> /■_ ■ ^ ^.'; A ? -F 1 right-hand side of equations f54) and (55). It is, however, conceivable that one of these components may be absent if the disturbing couple is due to some external cause. Thus if there is a couple round the x axis only, equations (54) and (55) become and Critical Speeds of Rotors. 155 — B?; — kv + Aw£= — 7(o 2 ±h\pcot, . . (59) Bf+tff + Ao^O, (60) giving the forced vibration 9 <> <> 2 V ~TT (c 2 2 -jpV) 2 -mV> 4 input. . . (61) sin The oscillatory motion is thus a whirl the nature of which depends on the speed. The whirl is a maximum when the denominator is zero, that is, at two speeds, one on each side o£ the stationary critical, and given by a ^ = ^— (63) There is thus a possible further second critical speed, corresponding to the + sign, lower than that already found corresponding to the — sign. For the ordinary rotating couple the direction of whirling is, of course, always in the direction of rotation, whether the speed is above or below the critical, and this is indicated by the fact that, as will be seen from equations (58), the sign of the amplitude in both planes changes, showing also that the phase of the motion has changed by 180°. But for an alternating couple about the x axis only, as will be seen from (61) and (62), the motion in the horizontal plane changes sign at each of the critical speeds indicated by (63), while the motion in the vertical plane has a further change of sign when p 2 co 2 = c 2 2 . It will be seen, if these changes are fol- lowed out, that the whirling is in one direction below the stationary second critical and in the opposite direction above that speed, while at the stationary second critical the motion is in the horizontal plane only, that is, at right angles to the applied torque. 11. The following example is given to illustrate the appli- cation of the above curves and formulse. A rotor consists of a solid cylinder 30 ins. diameter and 60 ins. long, running in bearings, 107 ins. between centres. From the deflexion diagram, the deflexion at the centre of gravity is, say, *0087 in. or '0221 cm. 156 Mr. P. Cormack on Harmonic Analysis of We then have : The first critical speed c x is -^L =2020 R.P.M. The ratio of the radii of gyration k 1 jk 2 is 6/19 = 0*316. Half the distance between bearing centres = 53*5 ins. The radius of gyration for the cross moment of inertia k 2 = 18-87 ins. The stationary second critical is, therefore, 6' 2 = Cl x ^=- = 5730 R.P.M. From the curves, fig. 6, it will be seen that the main oscillation will occur at 5720x1-21 = 6940 R.P.M. and that a double frequency vibration may possibly show itself at 5730 x 0-515 = 3120 R.P.M. Thus a turbo-generator designed for a speed of 3000 R.P.M. and having the above mechanical constants might show marked vibration on the overspeed test. X. Harmonic Analysis of Motion transmitted by Hookers Joint. By P. Cormack, A.B.C.Sc.L, Engineering Dept., Royal College of Science for Ireland *. 1. ^1|TITH the growth of high-speed machinery, the H determination of the accelerations of machine pieces becomes of increasing importance. These deter- minations are considerably simplified by expressing in the form of a Fourier Series the displacement of the piece under investigation. The value of this method in the analysis of the various phases of the motion of the mechanism of the direct-acting engine is well known. It is here proposed to investigate the coefficients of a Fourier Series for the angular displacement of the driven shaft of a Hookers Joint. The method being applicable to certain inversions of the slider crank chain, these are also included. The ease with which the coefficients can be determined, and calculations made from the resulting series, make the study of these mechanisms from this aspect one of considerable interest. * Communicated by the Author. Motion transmitted by Hooke's Joint, 157 2. In Hooke's Joint the point B moves in the great circle CBN and the point A in the great circle CAN (rig. 1). The arc AB is constant and equal to a quadrant of: the great circle. The point A receives its motion from an arm OA set at right angles to the driving shaft OX, while B transmits motion to the driven shaft OY. Let the angle between the shafts be y • this w r ill be the angle between the planes of CAN and CBN. In the spherical triangle ABC we have cos <• = cos a cos b + sin a sin b cos 7. Since c=7r/2, this becomes cos a cos Z> + sin a sin b cos y = 0. . . . (1) Plainly when B is at C, A will be at T; B will therefore move through the angle a while A moves through the anple &-7T/2. Writing <£ for a and 6 for b — tt/2, equation (1) becomes — cos (/> sin 6 + sin <f> cos cos y — 0. Put cos7 = (l — »)/(l + n), and we have (1 + n) cos </> sin 6 = (1 — n) sin cj> cos 6. .*. 11 (cos (/> sin 9 + sin (/> cos 6) = sin <£cos 6 — cos </> sin #. sin (<£ — 0) = n sin (</> + 6). 158 Mr. P. Cormack on Harmonic Analysis of Multiplying both sides by 6^ -0 ) gives e 2K<p~e)_ 1==ne 2i<i>_ ne -2ie t ... e 2i (t-^{l~ne 2i9 \ = l- ne -2i9 Mt-0) = (l-ne- 2ie )l{l-ne™). IT , 7T - and — y Since (p — lies between -f- and — ^, and n is less than unity, we may write 2i(cj>-6) = log (l-^- 2 *' )-log (l-ne 2i9 ) = - ne -2i9_±_ n 2 e -ii9_i n 2 e -QzO_ t < t + ng 2 ^ + \n-e^° -f- •fn 3 *? 6 *' + . . . = n. 2i sin 20+ ^n 2 . 2z'sin4(9 + §n 3 . 2i sin 6(9 + . . . ... <£-# = w sin 26 + \n 2 sin 4(9 + l?z 3 sin 60 + (2) It will be evident that (2) gives the displacement of the driven shaft relatively to the driving shaft. In practice the angle between shafts joined by a Hooke's coupling rarely exceeds 15°. Since cos a = r , we have J 1 + n n — (l--co3a)/(l + cosa) = tan 2 -. For a=15° we get w = '0173, so that we can without appreciable error neglect the terms containing the square and higher powers of n in (2) and put $-d = 7i sin 26 (3) For the above value of a, the maximum value of </> — 6 given by (3) is "0173 radian or nearly one degree. From (2) we have </> = + n sin 26 + \n 2 sin 4(9 + |n 3 sin 66 + (4) dfy = /^\ 2 ( _ 4?zsin 26-8n 2 sin4(9-12?i 3 sin 6(9-...). • • • W In obtaining (5) and (6) we assume the series formed by the term-by-term derivative of the member on the right in (4) and (5) to be convergent and to converge to the differential coefficient of the member on the left. In Motion transmitted by Hookes Joint. 159 obtaining (G) we take the case in which the speed of the driving shaft is uniform. From (5) the maximum speed of the driven shaft is readily seen to be co{l + 2rc/(l — ?*)} or w sec a ; the minimum speed is seen to be <w{l — 2n/(l-\-ii)} or « cos a, a) being the angular velocity of the driving shaft. When the angle between the shafts is not large, (G) may be written "(a)' w Thus the maximum angular acceleration of the driven shaft is very approximately 4?z&> 2 . If a = 15° and co = GO radians per sec, the maximum angular acceleration is almost 250 radians per sec. per second. Fig. 2. d *<t> • A t^- = — 4n sm dt 1 ' 3. In the mechanism of the oscillating cylinder engine, and the quick return crank and slotted-lever mechanism (fig. 2), we have sin <f> _ sin ty _ sin (# + <£) or sin (p = -sin (6 + (f>) = n sin (# + </>), where ?i= -. c v g2ty(l_ ne *e) = l_ w < /6> ,2/0 _ (l-ne- i9 )/(l-ne ie ), 160 Harmonic Analysis of Motion t; 2 Since <fi lies between +— and — - , and n<l, 2icf> = log (l_^-^)_log (1 -ne iQ ) = -ne- ie -±n 2 e- 2i9 - : k,nh- 2>i9 -... + ne ie + We 2ie + ^nh™ +'... = n . 2z sin + in 2 . 2z sin 20 + Jn 3 . 2i sin 3(9 + . . . . <£ = ttsin0 + in 2 sin20+§n 3 sin30 + (8) ^ = (?icos0 + n 2 cos20 + n 3 cos30 + ...)^. . • (9) ^ = (_» sin 0~2n 2 sin 20- 3n 3 sin 30-...) fc . . ! (10) R* 3. 4. In the Pin and Slot mechanism (fig. 3) we have "\|r = IT — — (£. - The angle <£ is given by (8), so that n ii ^ - rjr-O-n sin 0- T sin 20-^- sin 3(9- ..., (11) <i± _ _^ (1 + n C os (9 + 7i 2 cos 2(9 + n 3 cos 30 + ...), (12) ttt etc ^ = (~ V (n sin + 2n* sin 20 ■+ 3?* 3 sin 30+ ...). (13) It will generally be found that we need to consider but the first few terms of these series in making numerical calculations. L 161 j XL Short Electric Waves obtained by Valves. By E. W. B. Gill, M.A., B.Sc, Fellow of Merton College, Oxford, and J. H. Morrell, M.A., Magdalen College, Oxford *. 1. f INHERE have recently been discovered methods for the i generation of continuous oscillations of short wave length (of the order of about a metre) by means of three electrode valves. In January 1920, Barkhausen and Kurz f found that with hard valves — i. e., valves at extremely low pressure, if the filament and the plate were approximately the same potential, or, indeed, if the plate were at a potential considerably lower than the filament, provided that the grid was kept at a high potential with regard to them, continuous oscillations could be maintained in a circuit of the Lecher Wire type connected to the grid and plate. The wave length depended primarily on the grid voltage, but also on the emission from the filament and on the plate voltage. Whiddington J had previously described another method of getting oscillations of lower frequencies using a soft valve, i. e.j a valve containing gas at low but appreciable pressure. He employed more usual circuits for a valve, in that the plate was at a high positive potential with regard to the filament and the grid at a few volts above the filament. In this case longer waves were emitted, and he noticed that if V was the grid potential and \ the wave-length emitted, then X 2 V was constant §. There appear to be other arrangements not hitherto re- corded which will also give these waves. With a hard valve and with the grid at a positive potential, oscillations can be. obtained if the Lecher Wire system is connected across the filament and grid.; the plate may be positive, negative, or at the same potential as the filament, or it may be insulated. Further, the third electrode — the plate — is unnecessary, for oscillations can be sustained by means of a valve consisting of a filament and an anode formed as a spiral of wire con- centric with the filament, when these two are connected to the Lecher wires. An intermediate arrangement has been worked successfully in which the wave-length of the diode connected as above is modified by a cylinder concentric with * Co mm uni cited by Prof. J. S. Townsend, F.R.S. + Phifsikalischer ZeiUchriftj Jan. 1920. X AY biddington, ' Radio Review,' Nov. 1919. § For a general account of these experiments see ' Radio Review, June 1920. Phil Mag. S. 6. Vol. 44. No. 259. July 1922. M 162 Messrs. E. W. B. Gill and J. H. Morrell on Short the anode, but placed outside the valve and set at various potentials. The best conditions for these cases are still under investigation. 2. Barkhausen and Kurz were apparently unable to give any explanation of the way in which the oscillations were sustained, while Whiddington assumed that the emission of ions from the filament was discontinuous and occurred in bursts. The authors, on the other hand, do not think that any special assumptions are necessary, and that the ordinary conditions for the maintenance of oscillations by continuous emission will account for all the facts they have observed, provided that the time taken by the electrons to pass between the electrodes is taken into consideration, as this time is of the same order as the period of the short waves. In the present paper only oscillations of the Barkhausen type are considered in detail, but the theory can be extended to cover all the types, and an account of some experiments on the last type (with a diode) will be published later. It is worth noting that certain writers give the impression that the seat of the oscillations is in the gas or in the electrons in the valve, and that the Lecher wires connected to the valve serve only to demonstrate their existence *. It appears from our experiments that the wires or conductors attached to the electrodes are a necessary part of the oscillatory system. Even with the Lecher wires removed, there will always be some circuit composed of the connecting wires to the batteries or even the valve leads up from the sockets, which will have natural periods of a suitable order for short wave oscillation. This fact seems to have been overlooked in some recent determinations of ionizing poten- tials, where large emissions from a heated filament were used as a source of electrons. Oscillations will take place even when the valves contain a small amount of gas, but in all the experiments described in this paper gas-free valves were used. 3. It will probably be most convenient first to describe the experiments in detail, and then to set out the theory and apply it to the observed facts. Various valves were used, but mostly the Marconi M.T.5 valves, which were very kindly given to us by the Marconi Company. These valves consist of a straight filament FF held in the centre of the valve by springy arms. The ad- vantage of the spring is that when the filament is heated and * Whiddington's theory is independent of there being any external tuned circuit. Electric Waves obtained by Valves. 163 expand?, the spring prevents sagging. Surrounding the filament is a cylindrical wire grid, GG, composed of thin wire of square mesh, each square having a side of about 1*5 mm. The lead to the grid goes out at the bottom near the filament leads. A cylindrical plate, PP, surrounds the whole with its lead going out through the top of the bulb. These valves being used for transmitting purposes are very thoroughly "glowed out * and pumped to a very high vacuum. The filament emission is very high when heated with 6 volts direct, and for the low emissions that were generally used it was very constant. As the plate lead passes through the top of the bulb, instead of through the bottom and the sealed Fisr. 1. socket, very high insulation is obtained, and, if a strip of tinfoil connected to earth is placed round the outside of the gla-s, very small anode currents may be measured by an electrometer without any disturbance due to leakage. It is not necessary for ordinary wireless purposes that the valves should be constructed with the grid and plates either accurately circular in section or accurately centred with regard to the filament ; but for the purpose of calculation a svmmetrical system of electrodes is necessary and the M.T.5 valve used in most of the experiments was specially selected. All the numerical results to be quoted were obtained from this valve. There is no difficulty in getting the short-wave oscillations with many types of hard valve, the French type produces them quite easily, but the chief reason for selecting the Marconi M.T.5 type was that the electrostatic field between the square-mesh grid and the plate approximates M2 164 Messrs. E. W. B. Gill and J. H. Morrell on Short much more closely to the calculable field between two co- axial cylinders than does the field in the French type, wher^ the grid is a spiral coil of fine wire. The diameter o£ grid used was 1 cm. and that of the plate was 2*5 cm. to an accuracy of about 5 per cent. 4. The preliminary experiments were made with the apparatus arranged as in fig. 2. The valve is shown dia- grammatical ly : F is the filament, Gr the grid, P the plate, LL the Lecher wires, which were of copper wire each about 850 cm. long and spaced 5 cm. apart. They were suspended about 200 cm. above the floor from insulators secured to the walls at each end, and from one end were leads about 70 cm. long to the grid and plate respectively. The bridge consisted of two equal condensers, C, 0, joined through the heater-coil of a Paul thenno-j unction, T. The outer plates were fitted Fiff. 2. with contacts to slide along the Lecher wires. The capacity of these condensers is unimportant, provided it is large com- pared with the capacity of the valve. In practice, the capacities were of the order of 1 milli-microfarad. The terminals of the thermo-junction were connected to a gal- vanometer by two long leads, which are not shown. The sliding contacts were also connected to the negative side of the filament-heating battery B, that on the grid-wire through a high-tension battery V, and that on the plate-wire through a potentiometer S. which could raise the potential of the plate +6 volts above the negative end of the filament. Two sensitive milliammeters, A, A, gave the steady currents through the valve to the grid and filament respectively. A. rheostat, R, controlled the filament-heating. In all cases potentials are measured with regard to the negative end of the filament. electrons set free at the With this arrangement th< Electric Waves obtained by Valves.- 165 filament move outwards under the positive voltage, V, oL' the grid, and a certain number go direct to the grid and arc collected there, the remainder pass through the grid, and, if the potential of the plate is just less than that of the filament, they return to, and are finally collected on the grid. If, on the other hand, the plate potential is a little above that of the filament, a certain proportion of those getting through the grid reach the plate. If the plate potentiometer is now ad- justed till the plate current is just zero, and the bridge is moved along the wires, it will be found that with the bridge in certain regions a plate current appears. It was the appearance of this plate current which led Barkhausen to the discovery of the short waves. With the present apparatus these oscillations are also made apparent by the deflexion of the galvanometer attached to the thermo-j unction. The positions of the bridge at which the galvanometer gave a maximum deflexion were fairly sharply defined, and did not always coincide with the positions for maximum plate current. It is not necessary for the plate potential to be so adjusted that the plate current is just zero when oscillations are not occurring. The plate may be set at a considerable negative potential, or the plate voltage may be positive. It was found that for a given grid potential there is a certain plate potential at which the oscillating current through the thermo-j unction is a maximum. Also as the potential of the plate was increased, for plate potentials only slightly positive, if oscil- lations commence the plate current increases ; at a certain plate potential no change is noticed in the plate current ; and at higher potentials the plate current decreases. For the M.T.5 valve this critical potential was about 4- 2 volts, when the voltage drop down the filament due to the heating current was about 4 volts. In the first experiments with this apparatus the position of the bridge was varied and the current in the thermo-couple observed when the grid voltage V, the heating current, and the plate potential were all kept constant. The oscillating circuit consists of a condenser formed by the plate and grid of the valve, the distributed inductance and capacity of the Lecher wires up to the bridge, and the capacities C, C in series with the wires and with the short resistance of the thermal heater which connects them. Hence, if there is an optimum wave-length A corresponding to the grid voltnge V, and if, starting near the valve, the bridge is pushed along; the wires, maximum amplitude of oscillation 166 Messrs. E. W. B. Gill and J. H. Morrell on Short should occur when the above circuit is tuned to X, 2\, 3X, etc., these positions being indicated by the deflexions of the galvanometer connected to the thermo-j unction. Moreover, the distances measured along the wires between successive positions of maximum oscillations should be equal to - , and all therefore should be equal. It was soon found that this simplicity was not attained, in certain cases equi-spaced positions were found, but in the majority of cases there were at least two sets of positions forming two series of equal spaces, which, as the spacing distance of the two sets was different, appeared to indicate two optimum wave-lengths. These effects are due to the different modes of oscillation of the system, and, according to the theory which we give below, a grid voltage V will, under suitable conditions, sustain oscillations of short wave-length between certain limits. Any mode of oscillation corresponding to a wave-length between these limits will be maintained. It was therefore desirable to arrange the apparatus so as to avoid these complications. 5. The most obvious improvement was to give up the idea of finding the wave-lengths by moving the bridge, and to put the bridge and its leads at the far end of the parallel wires joined to the valve, and to measure the wave-lengths of the oscillations by means of a loosely coupled secondary circuit. The system of wires connected to the valve is thus fixed. A second pair of long Lecher wires were set up with a loop joining one end, and this loop was brought near the valve circuit. When the secondary is in tune with an oscillation in the primary the current in the primary is reduced. The deflexion of the galvanometer connected to the thermo- junction in the primary circuit may be reduced by 50 per cent, when the bridge in the secondary circuit is in the tuned position, and a movement of 0*5 cm. either way will restore the deflexion to its original value. The distances between the successive positions of the bridge on the secondary circuit, for which the deflexions of the galvanometer attached to the primary circuit are a minimum, are the same, and are equal to halt the wave-length of the oscillation in the primary circuit. All the wave-lengths quoted were measured on this form of wave-meter and may be taken as accurate to 0*5 per cent. * With the condenser bridge and thermo-couple at the far end of the Lecher wires the filament was heated to give an emission of a few milliamperes (this is low r heating for an * Townsend and Morrell, Phil. Mag. Aug. 1921, pp. 266-268. Electric Waves obtained by Valvesj 167 M.T.5 valve) and the grid voltage was raised by two volts at a time by means of batteries of small accumulators from 16 volts to 120 volts, while the plate was kept about 2 volts positive, as this gave large deflexions. The corresponding galvanometer deflexions are shown in fig. 3. The deflexions are plotted against grid volts ; as a thermal detector was being used, the deflexions are proportional to the mean square of the oscillating current. The curve shows that oscillations are occurring over nearly the whole range, but that there are maxima for certain voltages— viz., 16, 24, 42, 58, 82, 114, approximately. The wave-lengths measured as above give from 16 to just below 24 volts X586 cm., from 24 to 40 volts X451, and so on, the wave-lengths for successive portions of the curve being 366, 307, 262, 233. These correspond to the free oscillations of Fiff. 3. the system, the wires of which were 850 cm. long with leads to the valve about 70 cm. long, with a slight addition tor the leads within the valve itself. The system of wires connected to the valve therefore present a selection of various modes of oscillation with wave- lengths 586, 451, 366, 307, etc., cm., from which the valve chooses the one suitable for the particular voltage V between the grid and plate — the sharp rises just before the various maxima showing that the system oscillates on the longer wave-lengths by preference. For each particular wave- length there is a certain grid voltage which gives the strongest oscillations when the heating current in the filament and the plate voltage is constant ; but the heating current and the potential of the plate relative to the filament both affect the optimum voltage for a given wave-length. In- creased emission has the same effect, but this effect depends on the degree of saturation of the emission current. In the preceding experiments the wave-lengths of the oscillations were measured with a constant heating current 168 Messrs. E. W. B. Gill and J. H. Morrell on Short in the filament, but the current from the filament varied with the grid voltage. For the lower voltages all the electrons leaving the filament do not r^ach the grid space, some returning to the filament. For theoretical reasons it is more convenient to find the grid voltages which give the maximum amplitudes of oscillation on the various wave- lengths when the heating current is so adjusted that the same current flows from the filament to the grid space for all the voltages, the plate voltage being kept constant as before. The table below gives a set of experiments done under such conditions with an emission current of 6 milliamperes, and the plate at 1'3 volts positive to the filament. In column 1 are given the wave-lengths \ in cms., in column 2 the grid volts V, which excite these wave-lengths most strongly, and in column 3 the product X 2 V: — X. V. X 2 Y. 208 cm. 156-5 68xl0 5 233 122-5 66 262 92-5 64 307 68-5 ' 64-5 366 505 67-5 451 36-5 74 All these results, with the exception of the last, agree well with the relation X, 2 Y = const. It is not difficult to see why this agreement should be less exact as V decreases. The electrons concerned are not all moving under similar conditions. Owing to the voltage drop of the heating current down the filament, the field between filament and grid differs by about 4 volts for electrons starting from the extreme ends of the filament. And when V be- comes comparable to this 4 volts a disturbing factor is introduced. 6. These experiments thus give the grid voltages which produce the strongest oscillations on certain definite wave- lengths determined by the particular length of wire used. To find the range of wave-lengths maintained by a given grid voltage a slightly different apparatus (fig. 4) was used. An adjustable circuit was constructed of two rods, and two telescopic tubes fitted over the rods, so that the effective lengths of the system could be varied by sliding the tubes over the rods* The condensers and thermo-jnnctions were attached at the endsX, X 1 of the rods, and the ends Y, Y 1 of the tubes were connected to the plate and grid of the valve respectively, the other connexions being as before. Electric Waves obtained by Valves. 169 For brevity, the adjustable circuit will be referred to as the rods. It is not possible to graduate the rods in wave- lengths as against extension of the arms, as this wave-length Fig-. 4. depends on the emission and on the plate volts. Thus with a fixed length of the arms and 44 volts between grid and plate : (1) With plate potential fixed. Emission 2*2 m.a. \ = 311 cm. 6-8 306 9-8 300 (2) With emission constant at 5*2 m.a. Plate potential 1*2 volts. \ = 308 cm. 2-4 314 - Hence for a given setting X decreases as the emission rises, and increases as the plate voltage is increased. This is due to the fact that the plate and grid are not a potential node of the oscillating system, but are a variable distance from it depending on the alternating voltage necessary to sustain the oscillation, and this in turn depends on the emission and plate voltage. It is not, however, necessary to go further into this, as the wave-lengths were always found directly by a secondary circuit as in Paragraph 6, the rods being used as a convenient way of varying continuously the wave-length of tire system connected to the valve. With all the other factors fixed, the rods were pulled out a centimetre at a time and the oscillating current and wave-length recorded for each position. In one experiment the emission was 1*5 m.a.. the orrid potential 44 volts, the plate potential 1'8 volts, and oscillations were maintained from X = 320 cm. to X = 451cm. with a maximum oscillation about A, = 323 cm. It was always found that the maximum oscillation was close to the short-wave end of the rane^e. 170 Messrs. E. W. B. Gill and J. H. Morrell on Short The effect of: (A) varying the emission current keeping the grid and plate voltages constant, and (B) varying the plate voltage keeping the emission current and potential between grid and plate constant was investigated with this apparatus. In (A) increased emission broadened the range and de- creased the wave-length of maximum oscillation. For example, with V^ p = 44 volts and ~V p f—l'2 volts the wave-length for best oscillation with total emission 7*0 m.a. was 295 cm. With total emission 10'6 m.a. it was 274 cm. In (B) increase of plate voltage increased the wave-length and also broadened the range. Thus with Y^ = 44 volts, and total emission 3*8 m.a. with Y p f=l'2 volts X = 298 cm., and for V^ = 3*0 volts \ = 321 cm. This last observation must not be confused with the case in which the potential of the plate is increased and that of the grid kept constant. In that case also, increase of plate potential increases the length of the strongest wave, as was observed by Barkhausen, who attributed all the difference in wave-length to the alteration in potential difference between plate and grid. This cannot be the whole of the explanation, for, as stated above, similar results may be obtained by raising both plate and grid equally with respect to the filament. 7. A simple theory to account for the maintenance of the oscillations can be worked out by making some simplifying assumptions ; but a general theory will not be attempted, partly because the resistance of the oscillating circuits used was unknown and partly because if the assumptions are not made the calculations become extremely complicated. These assumptions are : — A. That the grid and plate can be regarded as forming a parallel plate condenser. B. That, of the electrons which leave the filament, a fixed small proportion pass through the grid in a uniform stream, and that each electron on passing through the grid has the same velocity. C. That the electrons which return to the grid from the plate side are nearly all collected directly on it, i. e., only a few pass through on the return journey. D. That the oscillating potential differences are small com- pared with the fixed potential differences employed. It is also assumed that the pressure of the gas inside the valve is so low that the number of collisions between electrons and gas molecules is negligible — this is certainly true for the valves used. Electric Waves obtained by Valves. 171 With these assumptions we shall only attempt to show that an oscillation can be maintained of about the right order of wave-length. The principle involved is the following : — Suppose the filament and plate are at zero potential and the grid at + V ; then the electrons from the filament which pass through the grid with a velocity v due to the potential V come to rest at the surface of the plate and return to the grid, which they again reach with velocity v. In the space between the grid and the plate the total work done by the fixed potential V on the electrons which move in this space is zero, all the work having been done between the filament and grid. If now superposed on the fixed potentials there is an alternating potential V s'mpt between grid and plate due to oscillations, the work done by the potential V sinp£ on the electrons is not necessarily zero. If the work is positive the electrons are abstracting energy from the oscillating system, and the average velocity with which the electrons hit the grid is increased ; the oscillations cannot in this case be sustained by the movement of the electrons. But if the work is negative the electrons are giving energy to the oscillating system, and if the rate at which this energy is given is at least equal to the rate of dissipation of energy in the oscil- latory circuit by resistance, radiation, or dielectric loss, the valve will maintain the oscillations. The average velocity with which the electrons hit the grid is in this case less than the velocity r due to the potential V which they acquire between fliament and grid, and hence the energy put into the system from the battery Y is not all used in heating the grid but part is turned into energy of oscillations. The above argument is not affected if, in consequence of the oscillation, some of the electrons are collected on the plate. In all cases, provided the total work dune per oscilla- tion by the alternating field is negative, an oscillation can be sustained if the dissipation of energy in the oscillatory circuit is not large. 8. The particular case in which the filament and the plate are at the same potential when there are no oscillations may be considered first. Let V be the potential above the plate of the grid and cl the distance between them. "When there are no oscillations the electron passes the grid with velocity v\ = \/ — V and is then subject to a constant retardation/, which brings it to rest just at the plate. If T 172 Messrs. E. W. B. Gill and J. H. Morrell on Short is the time the electron takes to pass from grid to plate, v = fT. A further interval of time T brings the electron back to the grid with velocity v. Assume now that superposed on the fixed potentials is an alternating potential V sin pt between plate and grid; the electric force due to this in the space between plate and grid y is ~ sin pt, and if — e is the charge on an electron the corre- spending force on it is — -—sin pt towards the plate. Since V is taken to be very, small compared with V the motion of the electron may to a first approximation be taken as determined solely by V, i. e., its time across is T and retardation /. The work done by V sinjt?£ depends upon the time t at which the electron passes the grid, and for a particular value of t the work is equal to l •— -psin ptdx. The axis of x being perpendicular to plate and grid and x = being on the grid. But the velocity at time t is JW-/(«-i„)=/T -/(*-<„), and the above work reduces to J fe — j (T -f t — t) sin pt dt, to (1) which finally gives ; Work on electron going from grid to plate = 2eY / r Y oospto sinpt — sinpT-ht \ similarly, the work done on the same electron as it returns from plate to grid comes out as -rpr{-cospt + 2L+ — C-s p2 —)■ • 12) Thus the velocities of the electrons on their arrival at the plate or on their return to the grid depends on t , that is, on the value of V sinp£ at the instant they pass through the grid. Assuming a constant stream of electrons through the grid, it is easily seen by integrating (1) for values of t between and — that the total work done per period is p Electric Waves obtained by Valves. 173 and similarly for (2). Hence, if all the electrons returned to the grid an oscillation would not be maintained. The possibility of a maintained oscillation depends in this case on the fact that in each oscillation a certain group of the electrons are collected on the plate and the integral of (2) does not in consequence include all the values of t between 2-7T and - and its value is not therefore zero, but may be P negative. The first step is, therefore, to find which electrons reach the plate. When there are no oscillations the electrons have sufficient energy on passing the grid to just take them to the plate against the potential V, and if therefore any extra work is done on them they will be collected on the plate, but if the work is negative they will fall shore of the plate and return to the grid. Expression (1) shows that all the electrons which pass through the grid at times t , such that Tcos/>^ sin pt — sinp(T + '. . . .,, , , ; + — —- 1 — is positive, will reach the • P l r plate, while chose for which it is negative just fail to reach the plate and return to the grid. Of the electrons then which pass the grid half go on to the plate and half return to the grid, the electrons running to the plate for a time equal to ir\p (half the periodic time of the oscillation) and then running back to the grid for time 7r/p and. so on. But the total work done by the oscillating potential on the two halves as they go from grid to plate is zero : and therefore the net work done is the work done on the return journey on the half which returns to the grid. To find therefore if an oscillation whose periodic time is 2irlp and amplitude V can be sustained by a grid voltage V it is necessary first to find the time T which the electrons take to pass Irom the grid to the plate under the field due to V alone, next to find from equation (1) the values of t Q for those electrons which return to the grid when the system is oscillating, and, finally, by taking the mean value of expression (2) for these values of t and, knowing the emission current, to find the total work done per second by the oscillating potential. If this work is negative and at least equal to the dissipation loss per second, the oscillation will be maintained. A table of calculated approximate results is given below for various values of the ratio T : l/p. The second column gives the values of pt for the electrons which return Values of pt for electrons which return to grid. Work. Y-4 165° to 345° negligible. 1040 cm. 150° „ 330° -•47 520 „ 135° „ 315° -•85 347 ., 120° „ 300° -•36 260 „ 90° „ 180° -•32 130 , 174 .Messrs. E. W. B. Gill and J. H. Morrell on Short to the grid in each oscillation, the third gives the total work done per second by the oscillating potential in arbitrary units for a fixed value of V and of emission, and the fourth gives the corresponding wave-lengths for the particular value, V = 44, for which T — 4*3 x 10"^ second (see next paragraph) : — r ~ 4T 7T 3;r 7T 2tt P = y In all these cases the work is negative and oscillations can be theoretically sustained, though in practice the dissipation losses in the oscillatory circuits are such that generally only those wave-lengths corresponding to the larger values of the work exist. For all values of the ratio T : ljp outside the range of the table the work is small. The larger the work the greater the amplitude of oscillation that will be sustained, and the periodic time 2ir/p of the oscillation of maximum amplitude for a given value of T is seen to be in the region of § T. In general, the wave-length corresponding to this will be sustained, and also a certain range of wave-lengths on both sides of it, the limits of the range being determined by the dissipation losses in the. oscillatory circuit. The theoretical result is in good agreement with several of the experimental results of paragraph 6. In the particular case recorded there for V = 44 the range of wave-lengths sustained was from 320 to 451 cm., with a maximum amplitude for 323 cm. Increase of emission broadened the range of wave-lengths sustained, which is in accordance with the fact that for a given V the work put into the oscillatory system is proportional to the emission current. The fact that the wave-length of maximum amplitude of oscillation was near to the short wave-length end of the range also agrees with the calculated fact that the Electric Waves obtained by Valves. 175 work done falls off much more rapidly on the short-wave side of the maximum than on the long-wave side. 9. To calculate the time T an electron from the filament takes to go from the grid to the plate when the grid potential is V volts above both filament and plate it is not necessary to assume the grid and plate to be parallel, but they may be taken, as they actually are, to be concentric cylinders of radii a, b. The retarding force on the electron when it is at distance v k b from the axis is -, where & = V/log e -. t a The equation of motion is therefore l 2 r - eh m which oives when integrated twice, remembering that dr = 0, when r = b, at \/jv 1o 4J V log T = b \ / -w log, - ] e~ x \lx. In the actual valve used a = '5 cm., 6 = 1*25 cm., and, taking -'= 5*3 x 10 17 E.S. units and measuring V in volts, 711 T = - r= second, w the accuracy of this being limited by the accuracy to which a and b are known and probably from 5 to 10 per cent. The wave-length for any relation between T and 1/p can now be at once calculated. If p = n7r/T, the time of one oscillation is 2ir/p or 2T/n and the wave-length in cm. is 6xl0 10 T/>?,. The simple theory shows that for the oscillations of maximum amplitude pT has a certain value about 37r/4. But Tec , andXx — , and hence the connexion between the v 7 V P grid voltage V and the wave-length X of maximum oscillation is X 2 V = constant. 10. The theory is thus in good general agreement with the experimental results, but there is one fact unaccounted for — that being the variation in the wave-length of the 176 Messrs. E. W. B. Grill and J. H. Morrell on Short oscillation o£ maximum amplitude, for a fixed potential between grid and plate, when either the emission is altered or the plate potential is slightly altered with respect to the filament. There is also a special case, which is forming the subject of a separate investigation, in which, when the plate is very negative (40 volts or so) with respect to the filament, oscillations can still be produced, but without any current reaching the plate at all. These oscillations are, however, very much weaker and more difficult to produce than those dealt with in our experiments. The simple theory which depends on the collection on the plate in each oscillation of a group of electrons will obviously not account for this special case. The explanation of the above considerations is to be looked for in the assumptions made in the simple theory. The first assumption that the grid and plate could be regarded as parallel is not important, as the field between cylinders of the size of the grid and plate used is not far from uniform. (It will be noted that the value of T was calculated for the valve used by taking the field between cylinders.) The only difference between cylinders and parallel plates on the simple theory would be to make the ratio T to 1/p for maximum oscillation slightly different. But the second assumption that the electrons pass the grid in a constant stream all having the same velocity requires more careful examination. This velocity is not actually the same for two reasons : A. Because there will be alternating potentials between the filament and the grid which set up a velocity distribution at the grid. B. Because of the voltage drop of the heating current down the filament. In the usual methods of producing oscillations by means of valves the alternating potential of (A) is most important, as it controls the whole action of the valve, but in our experiments it is only of secondary importance. The alternating potentials induced between filament and grid are smaller than those between grid and plate, and the major part of the work done by the alternating field on the electrons, which is what determines whether the electrons reach the plate or not, is done between grid and plate, and it is therefore nearly correct to say that all the electrons passing the grid at times t , such that expression (1) is positive, reach the plate. In the extreme case, however, when the plate is so Electric Waves obtained by Valves, 177 negative that the system is unable to oscillate by the method of: driving groups of electrons on to the plate, the oscillation is almost certainly due to a velocity distribution at the grid, as this means that the electrons do not pass the grid in a uniform stream, and allows the integral of expression (2) to be finite and not zero, as it normally is when all the electrons return to the grid. The comparative weakness of the oscillations in this case shows that the electrons have all nearly the same velocity when passing the grid. The simple theory should therefore be in agreement with the observed facts, as it is when the oscillations are mainly due to the collection by the plate, but as the plate is made more negative with respect to the filament the velocity dis- tribution at the grid becomes more important and the simple theory is less accurate. The velocity distribution at the grid will also bo affected by the emission, as this varies the space charge round the filament — this affecting the time the electrons take to pass from, the filament to grid, — and this in turn varies the small effect of the alternating field in this space. The effect of the voltage drop of about 4 volts down the filament is that, instead of dealing with one stream in the field due to the grid being charged to V volts, there are a series of streams moving under potentials varying from V to V — 4 (V being the potential difference between the grid and the negative end of the filament). The number of electrons in the various streams varies from a maximum number corresponding to V — 2, the middle of the filament being the hottest. The emission falls off equally on both sides of this middle point. In the general case, when the plate is slightly positive with regard to the negative end of the filament when there are no oscillations, some of the streams reach the plate and the remainder approach it closely, but to varying distances. If oscillations commence some of these latter streams are periodically diverted to the plate, while in the other half oscillations some of the former are diverted off. Thus all the streams concerned maintain the oscillation as in the simple theory, and unless Vis small the wave-lengths they each maintain best are nearly the same, so that the combined effect differs little from that of a single stream moving under potential V. The question of whether the mean plate current rises or falls when oscillations begin depends on whether the average Phil. Mag. S. 6. Vol. 44. No. 259. July 1922. N 178 Short Electric Waives obtained by Valves. density of the streams diverted to the plate exceeds or is less than that of those diverted from it. If the plate is at the same potential as the centre of the filament, no change should therefore occur. If it is above this the current should drop, and if below the current should rise, which is in agreement with the results in paragraph 4. In conclusion, we should like to express our thanks to Professor Townsend, who has assisted us with much valuable advice and criticism. Note on the Determination of Ionizing Potentials. In the experiments above described, the electric fields in the spaces between the grid, filament, and plate are similar to those used in experiments on the determination of critical potentials when a small quantity of gas is introduced. In the latter experiments variations in the plate current are observed as the grid potential is raised, and at certain potentials of the grid abnormal variations in the plate current are observed which are interpreted as indicating certain critical potentials, characteristic of the molecules of the gas. The experiments which are here described show that abnormal variations in the plate current are to be expected, due to oscillations which may be maintained where large currents of the order of a milliampere flow from the filament towards the grid when the gas is at a very low pressure. The effect of varying the potential of the grid is clearly shown by the curve of fig. 3. In all the ordinary methods of wiring the valve to the cells and galvanometers or electrometers, the system seems to be as often in a state of oscillation as not. Even if the plate is so negative that the oscillations are of the weaker type which do not affect the plate current, the difficulty arises that the oscillations superimpose an unknown potential difference between filament and grid, and the electrons are not moving under the fixed field alone. It is necessary therefore, in order to obtain reliable values of the critical potential, to take precautions to prevent oscillation, which is best done by using emission currents much less than a milliampere. [ !''•» J XTI. Pseudo- Regular Precession, By Sir G. Greenhill *. I^HIS is the gyroscopic motion described and illustrated in Klein-Sommerfeld's Kreisel-Tkeorie, p. 209, where a spinning top appears at first sight to be moving steadily in uniform precession at a constant angle with the vertical, but on closer scrutiny the axle is seen to be describing a crinkled curve of small loops or waves; so that in this penultimate state a realisation is obtained of a motion expressible by a function which does not possess a differential coefficient, a paradox fascinating to a certain school of pure mathe- maticians. A pseudo-regular precession, although invisible, would not feel impalpable to the analytical thumb passed over it, which would detect a roughness. But in regular precession the curve would feel quite smooth. In the investigation the axle may first be supposed moving in perfect steadiness with no trepidation or nutation ; and then to receive a small impulse, blow or couple, giving rise to the pseudo-regular precession visible to the eye. 1. Begin with a rapid spin about the axle, held fixed at a constant inclination, taken at first as horizontal for simplicity, in fig. 1. Fisr. I. For visible experimental illustration, it is convenient to take a large (52-iiuh) bicycle wheel, mounted on a spindle with ball bearings, and to prolong the spindle by sere wing- on a stalk, at one end or both. The end of the stalk may be supported on the hand and the wheel set in rotation by a swirl ; the hand accompanies the wheel in the precession ; or else the point may be placed in a cup fixed on the floor. When the axle is released from rest, it will start from a cusp and sink down, then rise up again in a regular series of loops or festoons ; so that to secure the uniform precession * Communicated by the Author. N 2 180 Sir G. Greenhill on an impulse couple must be applied, given by a horizontal tap of appropriate amount. The word moment or momentum is of such frequent occurrence in dynamical theory that we prefer to replace it by impulse instead of momentum, linear or rotational. Representing then the impulse (rotational) CR due to the rotation R about the axle by the vector OC, the impulse required to start the rotation from rest, or reversed to stop it again, the axle OG in steady motion will move round the vertical OG at a constant (horizontal) inclination, with pre- cession /j,, such that the vector velocity of K, the end of the resultant impulse OK, is equal to the impressed couple of gravity ; then CR{A=gM.h, M.h denoting the preponderance or first momeni about 0. This result, true accurately when the axle OC is hori- zontal (fig. 1), is obtained at any other inclination 6 of the axle with the vertical (nadir or zenith, figs. 2, 3) in the elementary Kindergarten treatment, where the top is supposed spinning so fast that the deviation is insensible of axial impulse OC and resultant impulse OK, and then the velocity of C may be equated to the gravity couple, making CR//, sin 0=gM.h sin 6, as before, when the axle was horizontal, on dividing out sin 6. 2. Hanging down inert, vertically from in fig. 2, the top forms a compound pendulum of S.E.P.L. (simple-equivalent- pendulum-length) 0P = £ = A/M/t, A denoting the trans- verse-diametral M.I. (moment of inertia) at 0, and as above the axial M.I. ; and in small invisible oscillation in a plane, the top will swing as a pendulum, and beat njir times a second, where n 2 =g/l, An 2 =gMh, or make a swing or beat in ir/n seconds ; MA may be called the preponderance about 0. Falling down from rest from" the upward vertical position, the top will have acquired the angular velocity co in the lowest position, such that by the Energy-Principle, iAo) 2 = %M/>, and the equivalent rotational impulse Aa) = 2A??, while An 2 =gMh is the equivalent of P in the Kreisel-Theorie. The impulse 2An = 2 ^(gMhA) is a dynamical constant of the top, and to a geometrical scale may be represented by a length Jc, in addition to the S.E.P.L. 0P = Z measured along OC ; and then if in any assigned top motion the constant impulse component G and CR, about the vertical OG and Pseudo-Regular Precession. 181 the axle OC, is represented to the same scale by d and D', a, a' = G, cr = m' /t ' 2An ' a in Darboux's notation (a different use of A from that employed above). Time can be reckoned in the pendulum beat, irjn seconds ; and the relation, CB. /n=gM.h = An 2 , can be written jji _ A n n ~CR ; or expressed in words, the number of beats per circuit of the axle is C/A times the number of revolutions of the top per double beat. The resultant impulse vector being OK, the component perpendicular to the axle, if horizontal, as in fig. 1, is A _ (?MAA _ A 2 n 2 ^"""CfT" CR' or to the geometrical scale, OC.CK = i& 2 , in the steady, regular precession. For brevity we are allowed to assume tacitly the geo- metrical scale, and to replace any dynamical quantity in an equation by its vector length, such as the axial impulse CR by the vector length OC, or a'. 3. To change this steady motion of the axle into a penultimate pseudo-regular precession, another impulse is applied about a vertical axis, supplied by a horizontal tap on the axle perpendicular to the plane OCK, in fig. 1. This will cause CK to grow to CK 3 , and the resultant impulse to change from OK to 0K 3 ; and to make the pseudo-regular precession advance through a series of cusps, we find that KK 3 = CK, and the axle rises from OC to OC 2 at an angle 2 with the upward vertical, zenith ; where C 2 reaches the level GK 3 of K 3 ; and here # 3 , the inclination in the lowest position, is \tt. By a general dynamical principle OK 3 2 -OK 2 2 =i* 2 (cos<9 2 -cos0 3 ) = 2OC.CK(cos0 2 -cos6> 3 ). For in the general unsteady motion of the axle of a top, where the inclination 6 is varying, a new component KH is added to the impulse OK perpendicular to the vertical plane 182 Sir G. Greenhill dO GOC, of magnitude Ay-, and then the resultant impulse do vector OH describes a curve in a horizontal plane GHK, with velocity equal to the gravity couple gMh sin 6. Fig. 2. N' The radial velocity of H in the horizontal plane is then -t— < = gWi sin 6 cos GHK = gWi sin 6-^ , and integrating, i GH 2 = ?MAA(E - cos 6) = A 2 n 2 (.E - cos 0) = iP(E— costf), iOH 2 = iA: 2 (F-cos^), to a geometrical scale, with E, F dynamical constants. The component HK is zero in the upper and lower position, where = 6 2 , Z . In the general unsteady motion of the top, the impulse vector moves from OK to OH with KH = Ay- ; and with GC 2 sin 2 0' OG 2 -2OG.OOcos0 + OC 2 sin 2 KH 2 =OH 2 -OK 2 = OH 2 (A^Y=2AV(F-cos0)- Pseudo- Reg alar Precession. 183 or with cos = :, and replacing OG, 00 for dynamical homogeneity by Darboux's 2A/j, 2AA', ^y=2;i 2 (F-^(l-^)-4(A 2 -2/Ji'cos(9+A' 2 )==2n 2 Z, thus defining z as an elliptic function of t. Resolved into factors, we write r L — z x — z. z 2 — z. z — z s , in the sequence oo >c x > 1 >z 2 >z> c 3 > — 1 ; and then z = z 2 sn 2 \mt + z* 6 en 2 ^mt, m 2 = 2n 2 (^ 1 — %). 4. Here with 00 horizontal in fig. 1, cos #3=0, OK 3 2 -00 2 2 =G 3 K s 2 -G 3 K 2 2 =CK 3 2 = 200 . OK. cos0 2 = 2CK, . OK, CK 3 = 20K. A greater impulse would make the cusps open out into loops in the pseudo-regular precession; but the cusps would be blunted into waves if the impulse was reduced. Reverse this tap, and K is brought back again to 0, and the axle would fall as at first from a cusp and rise again. In the first cusp motion where the axle rises to a series of cusps and sinks again to the horizontal, the motion is found to be pseudo-elliptic and can be expressed in a finite form, sin 6 exp (yjr — lit) i = \/(l — cos# ? cos#) +i >/(cos # 2 eos# — cos 2 0), connecting azimuth i/r with 0, the inclination to the zenith. The verification is left as an exercise. Here h = h' cos# 2 . In the second cusp motion, where the axle is horizontal and falls from a cusp, and then sinks down to an angle 6$ with the downward vertical, nadir, the (yjr, 0) motion is not pseudo-elliptic ; but azimuth yjr and hour angle <f> change place (0, (j>, ijr the Eulerian angles), and sin 6 exp (cft — h't) i= V (sec 6 3 — cos 3 . cos 6) + zV (cos # 3 — cos 6 . cos # + sec # 3 ), OR where Darboux's h'= ^— = i/(^ . sec S — cos 6 Z ) changes place with h, or d' with ~d ; h = h' cos # 2 and B now zero. But an interchange again of </>, ijr will give the (\jr, 6) motion of a non-spinning gyroscopic wheel, or spherical 184 Sir G. Greenhill on pendulum, with h', CR and ~d' zero, in which the axle is projected horizontally, with angular velocity 2 h = sin 2 $-—, and sinks down to an angle 3 with the nadir, rising up again to the horizontal, and this makes }i—n V{i£ . sec 6 3 — cos Z ). The motion can be shown with a plummet on a thread, say about 10 inches long, to beat as a pendulum twice a second, a double beat period of one second ; whirled round swiftly, the thread rising to the horizontal position, and sinking down again periodically. Then we find v 2 2 = U 2 h 2 = 2gl(sec #3— cos 3 ), v% 2 = 2gl sec S , and in the conical pendulum, at angle # 3 , v 2 = ^v 2 2 . The apsidal angle is found to be ^ = l7T+K i /(l-2« 2 ) -*7T(l-3 COS 2 8 ) as the plummet is whirled round faster. 5. But next suppose the axle OC is held at an angle 6 with the zenith, the wheel spun with impulse CR = OC, and then released, in fig. 3. The axle will start from a cusp, at 6 = 6 2 , and the motion in general is not expressible in finite terms as pseudo-elliptic ; but it will represent a gravity brachistochrone on a sphere. To make the axle move steadily at the inclination 6 with constant precession /*, the impulse vector CK is applied perpendicular to the axle, such that yu, sin 6 being the com- ponent rotation of the wheel about the axis OK' perpen- dicular to OC, CK = A/isin# (the inertia of the stalk being ignored), MO = Ayu,cos#, MK = A/*, KM drawn vertical to meet OC in M, with the condition GK . //,= gravity couple =gMh sin 6 — An 2 sin 0, 0M= GK : Arf sin 8 /jl then the geometrical relation CM-f MO = OC becomes A?? 2 Ayucostf-f — =CR, A 6 the condition for steady motion. Also OM.MC=AVcos<9 = fPcos<9, OM.MK==JP, so that K lies on this hyperbola with asymptotes OC, OG. Pseudo- Regular Precession. 185 Or the position of M is determined by drawing QQ' parallel to OC, to cut the circle on the diameter OC in Q,Q', when MQ 2 = ±Z: 2 cos0. This is for the small value of /-t in fig. 3, and the quiet precession when M is taken close to C, the other point Q' will determine a motion where the precession //, is swift and the motion violent. Fig. 3. OC, = 20M. MK' = 2MK. e Spinning upright with cos 0=1, fi-\ = -^ R ; and the fx A. two values of fi will give the independent normal invisible n 2 circling of the axle round the vertical ; there will be ___ rapid beats for one slow beat of the axle. P The general slight oscillation will be compounded of these two circlings, adding up to an epicyclic motion of the axle, a result obtained in this manner without any appeal to approximation. In fig. 2, the axle is pointing downward at *an angle 6 with the nadir ; here the slow precession of K is retrograde, but in the swift violent motion of the associated K', the precession is direct. 186 Sir G. Greenhill on 6. Next to make the axle rise to a cusp on d = 2 from = 6 Z in the penultimate pseudo-regular precession, the impulse applied is KK 3 , to make the axle come to rest on the horizontal G 2 K 3 in K 2 , C 2 , in fig. 3. Then in the general formula, or on the figure, with CD the perpendicular on K 3 M 3 , G 2 K 3 2 -G 2 K 2 2 = OK 3 2 -OK 2 2 = CK 3 2 -:P 2 (cos6> 2 -cos^ or CK 3 2 = K 3 D . K 3 M 3 =2 OM ,MK^, and producing MK double length to K', K 3 M 3 . 0C=20M . MK = OM . MK', implying that if OCLN' is the parallelogram on OC, the diagonal OL will cut MK' in L 3 such that ML 3 = M 3 K 3 , and K 3 is determined by drawing L 3 K 3 parallel to 00, cutting off the length CK 3 on OK, in fig. 3. 7. If the impulse is applied about the axle of the top, to increase 00 to 00 3 , and make the axle rise to a cusp in fig. 3, OC 8 .K 3 M 3 = 20M.MK, with MK = K 8 M„ so that OC 3 = 20M. Thus the axle will rise from the horizontal in fig. 1 to a cusp by the application of an axial impulse CC 3 = 0C. 8. The impulse might be applied about a vertical axis to the steady motion, making K rise vertically to K 3 ; and then in a cusped motion, with 00 changed to 0C 3 , and rising to 0C 2 at K 2 on the level of G 3 K 3 in fig. 4 ? G 3 K 3 2 -G 3 K> = OK 3 2 -OK 2 2 = C 3 K 3 2 =MK 3 . K 3 D = 20M . MK(cos0 2 -cos0 3 ) = 2OM . MK^ D , MK 8 . OG 3 = 20M . MK = 0M . MK', MK 3 _ MC 3 _ OM MK'~M0'~00 3 ' dropping the perpendiculars K 3 3 , K'C on 00 ; and drawing the circle on the diameter OC, with ordinate MQ', C 3 M . C 3 = 0M . M0' = MQ' 2 . Pseudo-Regular Precession . Then if L is the midpoint of OM, C 3 M . C 3 = LC 3 2 -LM 2 = LQ /2 , LC 3 = LQ', to determine C 3 . Fig. 4. 187 MK' = 2MK. MC = 2MC. L mid-point of OM. LC 3 = LQ'. 9. Applied about a horizontal axis in the vertical plane GOC, the impulse will make K move horizontally to K 3 , and the axle rise to a cusp at K 2 , C 2 , in fig. 4, if C 3 K 3 2 = M 3 K 3 . K 3 D = 2 0M .MK^, M 3 K 8 .OC 3 = 20M.MK. M,K,_FC, OM MK ~ FC ~ OC 3 ' FO, . C 3 0=20M . FC -2GK . KF = G / K' . K'F', thence a geometrical construction may be devised for the determination of C 3 and K 3 . 10. When the impulse is applied in the vertical plane GOC, the impulse vector starts out of the plane, from OK to OH, and K moves to H perpendicular to the plane GOC. 188 Sir G. GreenhiU on The axle then oscillates between 6 2 and # 3 , e,<e<e % , kh=aJ, GH 2 = KH 2 + GK 2 = KH 2 + ( OC-OGcos<?\ 2 CH 2 = KH 2 + CK 2 ^KH 2 + ( OG - 00 /°^ ) : \ sin / sin 6 J = 2OM.MK(E-cos0), f-OCcos<9\ 2 = 2OM.MK(D-cos0). Fig. 5. When the axle rises to a cusp, rises to 2 , K 2 at the level of G, where in fig. 5 KH = 0, OG-OCrcos0 2 = O, D=cos0 2 , OH 2 = 2 OM . MK (cos <9 2 - cos 6) = 20M . MK^5 = 2^ . KC 2 KH 2 = CH 2 - CK 2 = (2~ -i) CK 2 , giving KH the impulse. Pseudo- Regular Precession. 189 This caa be applied by hammering the rim o£ the bicycle wheel with a stick in a vertical blow at its highest point. Then OK3 2 -OK 2 2 = OK3 2 -OC3 2 = C 3 K3' 2 = 2 OM . MK (cos 2 -co8 3 ), M3K3 . K 3 D 3 = 20M . MK l -^ } M 3 K 3 . 00=20M . MK, and this determines the level of AM 3 in fig. 5, and provides a geometrical construction for the position of 3 . For 0M 3 . M 3 C 3 = OA sec 3 . M 3 K 3 cos 3 = OA . AG, so that, if B is taken in OG where AB = OA, 4 0A . AG = OG 2 -BG 2 = OK 2 2 -K 2 B 2 ; and the circle centre K 2 {in d radius K 2 B will cut off a length OE on OK 2 = OF = 20M 3 , and so determine the direction of OM 3 C 3 , the axle in the lowest position. When OC is horizontal, as in fig. 1, cos# 3 = 5 CH 2 = 2OC.CKcos0 2 = 2CK 2 , KH = KC, as at first ; and OC = OM, M 3 K 3 = 2MK, 9 2 + 6, = tt, the axle oscillating to an equal angle above and below the horizontal. 11. When the cusp motion is pseudo-regular and in small loops, it can be projected on the tangent cone of a sphere in a series of small hypocycloidal branches, and the motion is realised as discussed in the Principia, Book I, section X, when the tangent cone is developed into a plane and gravity radiates from a centre. With the axle horizontal, a necklace of brachistochrone cycloids is formed round the equator, with mean regular precession fi, fluctuating in azimuth between and 2fi, with azimuth interval 2^ = 2 Kk, tending as the rotation Rand axial impulse CR is increased to ttcos0 2 , in a zone above the equator of angular width ^it — # 2 , and area 2 7m 2 cos # 2 on a sphere of radius a, and the number of cycloids in the necklace would be about 2sec# 2 . 12. Even in the steadiest smoothest Regular Precession a close scrutiny will reveal under the slightest disturbance an almost invisible deviation from a perfect circular motion, in the shape of a progressive motion of an apse line, realised 190 Sir G. G-reenhill on easily with the thread and plummet; utilised by Newton to illustrate the Evection of the Moon, Returning to the general unsteady motion of a top in § 3, and its vector impulse OH, the velocity of H imparted by the gravity couple An 2 sin is horizontal and along KH perpendicular to the plane GOC. This velocity is the rate of growth of KH, added to the velocity of K carried round by the plane GOC with angular velocity ~- ; so that, with KO = A sin 6 -^, and putting -,, — Q, An 2 sin0 = A-^ + GK-^ =A-^+ . a , dt 2 at at A sin 6 3 . xl _ OC-OGcos0 rrn OG-OCcos<9 with GK = ; — z , KC = — — -. — ^ , sm 6 sin 6 , dGK KC dKG GK and dd "sinfl' dd sin0' obvious geometrically on fig. 3. Differentiating with respect to 0, with d0 = Qdt, 1 d 2 Q 2 a GK 2 -fKC 2 ~GK.KCcos0 = n? cos e-~ Q dt 2 ~ A 2 sin 2 GC 2 -3GK.KCcos<9 cos# A 2 sin 2 6 -( 2 , 30M . ON\ a OK 5 n 2 H r^ I COS #■ A 2 an exact equation. In a state of perfect Steady Motion of Regular Precession, Q and -7- are zero, and, in dynamical units, from § 5, OM.ON-AV, °M«i, gf-^-T' An />6 O.N /Z 2 Z' where X is the height of the equivalent conical pendulum, X=^ 2 = OA, l = g/n 2 = OY ; OC n u r-r— = - + - COS 6. An fM n OG An n A a = - COS04- ~, A 4 - W. GO 2 OK 2 sin 2 (9 An 2 A 2 n 2 in 2 6 /n 2 _ . , /j?\ . 9/i - T — = 2 4-2cos0+^ sm 2 0. ?r \/r nV Pseudo-Regular Precession. 191 Then in this Steady Motion, 7^ -y^T -fwi — 0, where m 2 OK 2 . . \ „ 2 AP 2 n 2= A^^ 4c0 ^ = 7- 2cOS ^ + X = OATOP^ £-2£ OA m 2 ~AP 2 UA * This result is exact and reached without any approxima- tion : and the slightest disturbance will give a nutation Q = Q cos {mt + e) , beating m/27r times a second, and the apsidal angle, from node to node, is Mr " 0P \ m AP In Darboux's representation of top-motion by a deformable articulated hyperboloid of the generating lines, the model is flattened into a rigid framework for Steady Motion ; and KM, KN produced to double length at S, S' will make the focal line SS' parallel to MN ; this will be revolved about the vertical line ON with constant angular velocity jul. The small nutation will be due to a slight play or baeklash in the frame. 13. The same argument can be applied to the invisible oscillation of a Simple or Spherical Pendulum, or to the apsidal angle of a particle describing a horizontal circle on a smooth surface of revolution about a vertical axis. Taken as the axis Oy, the general equations of motion of the particle are 1 dx 2 Idy 2 1 „d^ 2 , „ , . and x 2 -j- = K (impulse) ; d v d*\l/* so that, with -=-' =Q, and eliminating -j- , and differentiating with respect to #, dx = Qdt, Q C -~ = ~, dt\ 1+ dx 2 ) + ^ dx dx 2 x s +9 dx~ V > 1 d^ ( djf\ dQdylij td 2 y\ 2 Q dt 2 \ + dx 2 ) + dt dx dx 2 "*■ H \dx 2 ) ~dy d 2 y , 3K 2 d 2 y A dx dx z x* J dor exact equations. 192 Pseudo-Regular Precession. In a state of Steady Motion in a horizontal circle, Q = 0, ^=0, K=^ 2 , g = V?x d £- =V . NG, v*=^^g® d £ =g . NY? if the normal and tangent at P meet the axis Oy in G and V; and then with -f- = tan 6, ax oscillates between close limits #±/3, Q— (ot+ft) sin 2 \mt + {*—/3) cos 2 fynt, Q = m/3 sin mt ; and the particle beats mj2ir nutations per second, syn- chronizing with the beat of a simple pendulum of length \—g\m 2 , where A ~ VNG + dx 2 ) co * GV + PR' where PR is the semi-vertical chord of curvature upward of the profile curve of the surface. Then on a cone X = JGV ; and on a sphere or spherical pendulum, PR = NG, and to radius a, - =3 cos 6 + sec 6. . A, For a profile given by y = cx 7 \ d -l=n^, NG = ,^ = -, v 2 = 9 .NV = w, dx x dy ny * J J dx 2 x 2 dx 2 n + 2 Thus on the surface of a free vortex, where the angular impulse xv is constant of all annular elements of liquid of the same volume, x 2 y is constant, and n— — 2, X = co . On a motor or bicycle track of this shape, the steering will be easy, and a change of place can be made without difficulty or danger, as with the annular elements of liquids in the vortex volume. A start is made with moderate velocity from the circum- ference of the track where the slope is slight, and the car The Binding of Electrons by Atoms. 193 is steered with increasing velocity down towards the middle, where the cars can pass and repass without difficulty. To avoid a deep hole in the sink in the middle, the profile can change to the parabola of a forced vortex, where v= M , n = 2, y=^-, NG=p, A.=iGV = iSP. On a horizontal circle of this track of one lap to the mile, NP = 840 feet ; described in two minutes at 30 miles an NP hour, NV=60 feet, and cot 0= ^= = 14, a slope of 4°. Raise the speed to 60 miles an hour on this track, NP = 420, NV = 270, feet, and the slope is nearly 30°, the circuit of two laps to the mile made in 30 seconds. At a speed limit of 90 miles an hour, NP = 280, NV = 540, feet; round a circle of three laps to the mile, on a slope of over 62°. The surface could then change to a paraboloid, with a flat area in the middle, where a car could come to rest. XIII. The Binding of Electrons by Atoms. By J. W. Nicholson, F.R.S., Fellow of Balliol College, Oxford*. ACCORDING to the quantum theory of atomic structure and of the emission of line spectra, the paths of the electron in the atom vary according to the particular co- ordinates used in the process of quantizing the separate momenta. Thus in the simple case of a hydrogen atom, containing a nucleus and one electron, we may use either spherical polar or parabolic coordinates, and the admissible orbits are entirely different in the two cases. Yet the final values of the atomic energy are the same, and consequently each method yields the same theoretical spectrum. It has been suggested that there is in fact, in every case, only one type of coordinates which can be used, when all the modi- fying circumstances, such as the variation of the mass of the electron with speed, are taken into account. The only pro- blems yet solved are those in which the separation of variables, after the manner of Jacobi, can be effected, and the contention is in fact that there is, in every case, only one set of coordinates which allows this separation, when non- degenerate cases of the motion are discussed. But it is generally believed that the atomic energy is in all cases determinate and definite. We shall show, in the * Communicated by the Author. Phil. Mag. Ser. 6. Vol. 44. No. 259. July 1922. O 194 Dr. J. W. Nicholson on the first place, that this conclusion requires modification when the path extends to infinity. The hyperbolic orbits of Epstein, which have been used extensively in the inter- pretation of certain groups of 7 rays associated with many of the chemical atoms, constitute an instance, and we shall show that they rest on a mathematical error, and that in fact it is not possible to preserve finite phase-integrals in the process of quantizing the momenta. In fact, it appears that the whole process is only applicable to finite paths, and gives no clue to the phenomena taking place during the binding of an electron which comes from a considerable distance. In another form, the question we propose is as to whether a hyperbolic path is possible in the same way as an elliptic one. Such would, of course, be characterized by a positive energy W. Certain available evidence of a simple kind, apparently not hitherto noticed, is in existence. For the existence of such paths involves the existence of parabolic paths, with W = 0. In passage from a stationary state of energy Wi (negative) to a parabolic path taking the electron outside the atom altogether, a quantity of energy Wi should be involved. Spectral lines given by hv = W Hi where W n corresponds to any one of the stationary states, should thus exist. In other words, the ' limits ' of spectral series should themselves be spectral lines. But there are two reasons why evidence on these lines cannot be decisive, especially when it is negative evidence. For in the first place, the values of W n determining the limits of series are of such magnitude that only for two or three, in any case, can the corresponding lines come into the visible spectrum, and with only hydrogen atoms and charged helium atoms to test, and enormous band spectra for both elements, the test cannot readily be applied. Moreover, the probability of an electron entering the atom in a parabolic rather than a hyperbolic path is so small that any resulting lines could hardly be expected to be of visible intensity under ordinary conditions. We consider, therefore, that the question whether limits of series are themselves spectral lines, on the principles of the quantum theory, cannot, at least at this juncture, be examined in the light of experiment, and that it must remain a matter of deduction from other phenomena. We find it necessary, as stated, to disagree with the hypothesis, explicitly indicated several times by Sommerfeld and others, and implicitly assumed at least by the remaining Binding of Electrons by .Atoms. 195 writers on tlie quantum theory of spectra, that the energy W is always completely determinate when all the momenta are quantized. This can be disproved not only for fictitious laws of force in an atom, but for laws which must actually occur in systems with an existence, if only a temporary one. Consider, for example, a simple doublet and an electron in orbital motion about it. Regarding the doublet as stationary, and of moment M, its external potential is M cos ~ r* when it is situated at the origin, with its axis along the axis of z, using spherical polar coordinates. The equation of energy for an electron moving in its presence is im {r 2 + r 2 fl 2 + r 2 sin 8 6 j> 2 } 4- M * C 8 ° S ° = - W. The momenta are, in the usual notation, BT . BT %d Pi = 57. = mr > Pi = ^ = mr .°> Jh — ;— r- — mv 2 sin 2 0$, 09 so that Me cos 6 I r r L siir V J r l = -w. Xow <f> is a speed coordinate as usual, so that p z == const. = n-Jifeir when subjected to the quantum relation: n x being an integer. For the Jacobi solution, we must also take, in separating variables, P2 2 + ~P\a + 2mMe cos e =/3 £ suv 6 ^ where /3 is constant, and >{pi 2 +^} = -w. Thus -2mW-J. 02 196 Dr. J. W. Nicholson on the With a positive W, the motion is not real. Thus W must he negative and the path necessarily extends to infinity. A critical value of r is \J _ w , and the other is infinity. The phase-integral for p 1 is pi dr *J v -2mW which is infinite, but nevertheless independent of W. For writing it becomes 2m W = 2/^(1-^ A finite integral is secured, — Epstein's procedure, Joy instance,— by. using the phase-integral not for p r , but for Pi~~ {pi)r=x, which in the same way yields again independent of W. Now ft is quantized, or expressed definitely in terms of integers already, from the phase- integral for the momentum p 2 . The phase integral for pi can only, in this case, lead to another expression of similar type for j3 t but to no expression for W. It is not at all clear that the two expressions for f3, also, can both be valid simultaneously. This possibility has hitherto apparently been overlooked by authors in this subject. No case has, however, been noticed in which W is inde- terminate for a finite path. One very important conclusion is that the whole investigation is valid for a negatively charged atom with a distant electron. We proceed now to discuss the possible existence of definite paths with a positive total energy and infinite extent, for a single electron around a nucleus of charge ve, situated at the origin. This is Epstein's problem, which he treats as only two-dimensional. The energy equation is •Z y, 2 C r r z sin- 6 ) r Binding of Electrons by Atoms. 197 where W is positive, and represents the total energy, and the p's are the momenta. We have thus p. d = const. = -i-, Lit sin 2 0' sin 2 being clearly positive, The phase-integral for p 2 is nji = \p, d0 = 2\ d0 . \J /3 2 - the limits being the suitable values of for which ^ 2 = 0. The factor 2 represents the double journey in this co- ordinate, sin^=^, where ifr is one of the limits, and the other admissible value, for a real integral, is 7r — -v/r. Thus n 2 h = 2^ d0 V£^3~7sin 2 Write P 2 w = sm^ &) + — o cos w > 2?3 and we have n 2 h A Q /3 2 — ^ 3 2 f ff/2 COS 2 0) di&> — ^P 2 fi>2 ? ^ 3 -° sin 2 o)+- 2 cos 2 o) ^3 2 or with tan co = t, n q h = -h; ft J. (1+^(1^) = 4 J 8(tan- 1 *-§tan- 1 %'l 198 Dr. J. W. Nicholson on the whence £ = 27T + p 3 = v 1 ^ 71 *)^' these integers being thus additive, in the usual way. The phase-integral for p 1 is j dr^/2 n 3 k = \ dr\/ 2mW + /3 2 if we seek to quantize ^ as it stands. The limits would then be a positive value of r and infinity, for half the path, and the integral would be infinite. But it is clearly necessary to suppose that when the electron is at infinity, out of range of action of the nucleus, it should not be subject to a quantum relation, so that (j»i) r =« is not affected by the rule, and only the variable part P1-O1X0 is so affected. Yet this question of quantizing p ± presents some difficulties in whatever way it is suggested that it should be effected, and we consider that Epstein's discussion of the matter is very incomplete and not logically justifiable in its mathematical procedure. We shall thus consider various alternatives which may give a finite phase-integral. Now the actual r-path is not a passage from r = oc (say) to r = ao and back, and the phase-integral is not twice the defiuite integral between these limits. The electron goes from a limiting radius to infinity, and back to the same radius elsewhere, and the passage through infinity distin- guishes this phase-integral from those which occur in the other coordinates. We must, of course, also remember that the sign of p x depends upon the part of the path concerned, — whether the electron is departing or returning. The critical value of r is the positive root of 1 _ mve 2 + \/m 2 v 2 e 4 +2mWj3 2 _ 1 r ~{P~ ~ ~u Writing, generally, with a new variable <f> 1 _ mve 2 _ AiiW + 2 r ~W~ ~ V ~gi — lilt/ c. I 'V I' 1, v £■ l^ —II' YT kj JL . r Jgi = ~ a ( sa ?> 1 mve* /m 2 v 2 e± + 2mW6 2 V & ~ cos 9> Binding of Electrons hy Atoms. 199 we have $ = in the critical position (perihelion, in the usual terminology), and mve 2 1 /m 2 v 2 e 4 + 2mW8 2 cos< t>=--pr/'\/ ' —p- -=cos77 (say), when r=oo. What is required for the correct evaluation of the phase- integral is a continuous variable which shall change in one direction, — and thus give a definite integral, — as r goes to infinity and returns, the sign o£ p x being automatically taken into account, — or the sign of pi— (piXc when (pi) m is not zero as in a parabolic path. The new variable cj> has this property, and ranges from zero to 2tt as r goes through its changes. We have denoted its value, when r=co, by r) above, where rj is evidently an obtuse angle. The phase-integral for jt^ alone would be r2 V n 3 h=\ xrr 2mve 2 B" 1 dr \/ 2mW H S; (the square root being properly interpreted in different regions) where - = ~^r -f- 7^ V m Ve 4 + 2m W/3 2 cos # r (3 1 p z = ™^ + ~ VmW + 2mW/3 2 cos 77, and we find dr= T . *? + ** - v , £, p VZmW + mVeS (cos 9 — cost;) 4 // V n TTr 2vme 2 /3 2 q . , 2m W + ^ = £ sin <£. r r p. • If the integration were continuous throughout, — as as- sumed by Epstein, — we should thus have n z h= j3 \ sin 2 (f> d(j> J (cos<£— cost;) 2 sin 2 $ d$ (cos<£— cosr?) 2 ' which is an infinite integral, as would be expected. 200 Dr. J. W Nicholson on the If we merely quantized over the finite part o£ the hyper- bola, — another possible suggestion, — we should have n s k= fi ( f " + f * ) ,. *}»'* v d4> 6 I Jo } 2n-n) (cos </> -cost;) 2 r =4' sin 2 </> dcf) (COS <j>— COST;) 5 which is again infinite. The nature of the first infinity merits a remark, however, for it is independent of rj and therefore of W. For d<f> cos?; j_C* sin 2 <t>d<l> r sin </> ~| __ C n cos <$> J (cosc/>— cos*;) 2 — |_cos<£— cost;J J cos<£ — r sin<i "1 C* d<b = \ — — — it— cost;! -r-t- . |_cos <p — cos 7] J f cos (p— cost; The principal value of the last integral is well known to be zero, for all values of t;, so that the last term is zero. Our equation would be ^=-2^8+ r f* i, Lcoscp— cost; J where the principal value of the bracket must be taken, i. e. it is to be interpreted as Lu { r :»+ r v r f» r }. I Lcos — cos t; J LCOS0— COS^Jjj+e J This becomes ^ C e sin t; e sin t; J 2e which, though infinite, is an infinity independent of rj and therefore of W. We have another aspect of the indeter- niinateness of W for such paths. Our fundamental objection to Epstein's mode of integration may now be introduced. He integrates />i— (pO^? an d not pi, but this fact does not affect the question. For as (/> ranges between and 27r, if p 1 — /(<£), we have p ] varying con- tinuously with 6, and remaining positive, till 6 — r\. Then/?! becomes —f(<f>i) when <f> = 27r — ^>i on the return journey after <£ = 27r— -t;. Between (/>=?; and <£ = 27r — t;, the value of r should be infinite, and p x changes from V^mW to — v 7 2wiW, as in the figure. Binding of Electrons by Atoms. 201 The variation of p x between + v/2mW at infinity is the source of trouble, and it takes place while - =0. p, = - VJmW e = 277—77 Epstein takes twice the integral from </> = to <£=7r, but according to the substitution formula, r is negative when <£ goes frpm 77 to 77-, and negative values of r are clearly not permissible. A suitable integration for the infinite region cannot in fact be effected, and any supposition of a suitable variable in place of <£>, for the change of p 1 at 00 from V2mW to — \Z2mW, would be entirely arbitrary, — but as it could not lead to a finite phase-integral, we pursue the matter no further. These considerations, nevertheless, have considerable force when, thrown back as we now are upon the necessity, if the quantum theory is applicable, of using p l — (pi) mJ we attempt to quantize this. We have, when # = 77 k>i).= ^2mW = -* sin 77, where g= \/m 2 i>V + 2m W/3 2 as before. And when </> = 27r — 77, • (#)•= £ sin (27T-77) = - I sin 77. From </> = to $=77, Pi - (Pi)oo = I ( sin 9 - sin f) • From <f) = 27r — n to </> = 27r, Pi - (Pi)* = + I ( sin * + sin ^ and from (p = r) to $ = 27r — ?7, 202 Dr. J. W. Nicholson on the With the value of dr, the phase-integral n 3 h=\ dr( Pl -{p,)J Je=o breaks into three parts, thus Jo . (cos </>- cost;) 2 ^J„ Y T 271 " sin (ft (sin </> + sin nMcfr J 27r _„ (cos</>- cost;) 2 Jo (cos</>~ cost;) 2 ^J „ by a simple transformation. Finally, the only accurate phase-integral is f' sin»(»in»-Bi n ^)^ ,.J (cos 9— cost;)"' while Epstein gives, in our notation, nji — 28 f "" sm< ^ f s i K< fr~ sin?;) ~ J (cos <j>— cost;) 2 the part of his range from r/ to it involving a meaningless negative value of r, and violating p 1 = (p 1 ) x though the moving electron is at infinity. The principal value of Epstein's integral is, using the indefinite integral for the function in the form, readily obtained by parts, J sin $ (sin ff>— sin 7;) (cos <£-- cost;) 2 ™ sm 0— sin 7} , , , 1 2 = 7 '- —6+ COt T) . lOge S T cos </> — cos t; r n I . rj+(p of the type ^ = 2^1-1^—1—1 I 7rsmT;J or 2 _ (wi + rc 2 + rc 3 ) 7r sin rj n 1 -\-n 2 ' and ultimately 2 meV J 1 w= ^ 7J "" / . . \2 / — (Tl! + W 2 + %) - (^1 + W ? / — generalized from his value which relates only to a plane Binding of Electrons by Atoms. 203 hyperbola. We have the sum n\ + n 2 of the angular quanta in place of his single integer. But this formula, with all the applications he makes to characteristic 7 radiation, is not tenable, as resting on a mathematical error. Its apparent success appeared at one time to the writer to justify it as an empirical formula, in spite of his independent investigation, outlined above, indi- cating the impossibility of quantizing such orbits. Close examination, however, of the calculations of 7 radiation and so forth made it clear that they' were in several cases illusory, and determined more by order of magnitude than by the nature of the formula. There is one convincing argument against the formula, however. It should give an emission spectrum for all values of W], n 2 , n s and wi 1: m 2 , m 3 making — W(m l9 m 2 , m. d ) +W(n 1 , n 2 , n 5 ) positive. This can be tested in great numerical detail on the spectrum of a hydrogen atom, and the test fails entirely. No spectrum line is found, — in the secondary hydrogen spectrum, — in any of the assigned positions. Thus the formula really fails as an empirical one. We have seen above that it must be replaced by J (cose/)- cos 77) 2 •2/5 m 6— sin 79 , , ■ , J , — - — <P + C0t 77 l0£ e < - cos 6 — cos 77 r &e ) . - sin which is logarithmically infinite. The attempt to obtain a finite phase-integral, in this manner, in fact fails, and we must give up the hypothesis that even the variable part of p 1 can be quantized for the infinite path. It is not difficult to see that this conclusion is general for an} r infinite path which is possible for an electron about a physically existent atom, whose nucleus can always be regarded, for the present purpose, as a superposition of free charges and a set of doublets. We have demonstrated the result for a single free charge, and previously for sets of doublets. Further analysis of the more general case does not seem necessary, and could readily be supplied by the reader. Our conclusion must be as follows : — A determinate and finite value of W cannot be obtained for an electron moving about any atomic nucleus, if the path involved takes the electron to infinity. L 204 ] XIV. Theoretical Aspects of the Neon Spectrum. By Laurence St. C. Broughall *. T^HE object of this paper is to attempt to explain the spectrum o£ neon in a manner somewhat similar to that used by Bohr f in his explanation o£ the reason for the existence of the Balmer series in the hydrogen spectrum. The principle on which this hypothesis rests is that when an electron rotates in a fixed orbit it does not radiate energy, although the principles of electrodynamics state that it should ; if, however, the electron changes from one orbit to another, then energy is emitted, provided that the kinetic energy of the electron is less in the second orbit than in the first. In order to account for the spectrum, it is assumed that the energy emitted is numerically equal to the product of the frequency of the spectral line produced and the quantum constant. We thus obtain the equation E = nA, where E = energy emitted, n = frequency of the resulting radiation, and h = quantum constant. In the case of hydrogen, it was assumed that the orbit of the electron was circular, and then the attractive force between nucleus and the electron due to their equal and opposite charges was balanced by the centrifugal force of the electron due to its rotation about an axis passing through the nucleus. The energy of the electron can thus be found for any radius of orbit. When the electron changes its orbit, it moves to one with a radius which is an exact multiple of the radius of the original orbit. In this manner the change of energy due to a change of orbit can be found, and then, using the equation given above, it was shown by Bohr how the constant of the Balmer series could be found ; and the value so obtained agreed extremely well with that found by experiment. In the case of neon, we are dealing with an atom which contains more than one electron ; and since the atomic number is 10, it follows that if the atom is to be neutral, then there must be 10 electrons present to annul the excess of 10 positive charges in the nucleus. * Communicated by the Author. t Phil. Mag. vol. xxix. p. 332. Theoretical Aspects of the Neon Spectrum. 205 In order, therefore, to study the atom, it is. essential that the electrons be given definite position relative to one another. This has been undertaken by Langmuir *, and there is considerable evidence in favour of the postulate that eight of the electrons arrange themselves at the corners of a cube at the centre of which the nucleus is situated. The other two electrons are imagined to lie within this cube, probably on a line joining the mid-points of any pair of opposite sides. If we make use of this hypothesis, and further are in possession of data which will allow us to find the length of the diagonal of the electron cube, then it was shown by the author f that it is possible to calculate the angular velocities of the electrons about the nucleus. Siuce the determination of the spectral lines is an ex- tension of the matter given in that paper, it will be advisable here to state the principles on which the calculations of the electron frequencies depend. It has already been stated that the two inner electrons will probably lie on a line joining the mid-points of any pair of opposite sides. If this be the case, then the electrical forces acting on the outer electrons due to the other electrons in the outer shell, and to the two inner electrons, will be the same whichever electron we take, provided that the two inner electrons are equidistant from and on opposite sides of the nucleus. The next consideration was the axes of revolution. As before, it was desired if possible to get the forces acting on the outer electrons due to centrifugal action the same for all of the electrons. If we take as axes the three lines which pass through the mid-points of the three opposite pairs of sides of the electron cube respectively, then the above con- dition will be satisfied. In the diagram the axes of revolution are illustrated by XX', YY', and ZZ'. The inner electrons being on the axis XX' will only rotate about two axes. It is, of course, quite immaterial which axis the inner electrons lie upon. The forces acting on any outer electron were then considered, and were taken along the three sides of the cube which meet at the point where the electron is situated. Now, since the electron must be in equilibrium, so the force along each of these lines due to electric;) 1 attraction and repulsion and also due to the motion in a circular orbit must be equal to zero. * General Electric Review, 1919. t Phil. Majr. Feb. 1922. 206 Mr. L. St. C. Broughall on Theoretical In this manner three equations were obtained, namely 39e 2 l e 2 l 775 + e 2 (r + l) e 2 (r — l) U 2 T 2s 3 T [(r + /) 2 + 5 2 ] 8 ^ [( r -Z) s + jj s ] 3/2 + m(o 1 2 l + m(o 2 2 l, 390 s / 2 £ 2 Z = 77* + oTa + £ 2 Z + 4c 3 ~4/ 2 ^2.9 3 ^ [(r + Z) 2 + s 2 ] 3/2 ^ [( r -Z)* + iS]3/* + mo^ 2 / + mcofl, . . . J Z e 2 Z e 2 l Me 2 l + ±c z ~4Z 2 " r 2.5 3 ^ [(r + 2 + 5 2 ] 3 / 2 " r [(r-l') 2 +s 2 ] + 171(0% I + 11l(0 2 l. . . Fig. 1. Y l£ - 2l — »e Q c 5 i\ \ \ e^ ~ ~~~~2 r=^ -~-~~J____\. i = ^ r ' c i i i l\^io i E i \ e 1 >?9. J l\ *4 e 3 e 2 -z ! X 1 (I.) (II.) (III.) Y 1 Another equation can be obtained by considering the forces acting on either of the inner electrons along the line joining the two inner electrons. Equating the forces to zero, we found that 10e s 4<? 2 (r + I) » + ±e 2 (r-l) r 2 ' [ (r + I) 2 + s 2 ] 3 / 2 T [ (r - /) 2 + s 2 1 *' 2 + 4r 2 -f- m(Oi 2 r + m(o 2 2 r, . . (IV.) In the above equations, e = charge on an electron ; Aspects of the Is eon Spectrum, 207 r = radius of orbit described by the two inner electrons ; coi = angular velocity about YY' ; co 2 = angular velocity about ZZ' ; &> 3 = angular velocity about XX'. I = I length side of electron cube ; s = ^ surface diagonal of the electron cube; c = ^ diagonal of cube, 'a' and i V are of course functions of ' c,' and if the latter is known, then ' 5 ' and ' I ' can be found, m — mass of an electron when its velocity is small compared with that of light. * From these equations it was shown by the author that an equation involving only ' r,' ' c/ ' /,' and ' s ' could be found. In the paper mentioned, '/' and ' s' were not expressed as functions of ' c,' but expressing them as such, since 1 = — — and s= — ~— , we obtain the equation o o 4-34 r-h'Dllc f, 2-308c-) r + 'Dile r . Z'dVSc 1 " L0'+'577c) 2 + -667c 2 ;p ) r J r-'577c f 2'308c^ 5^ [(r--577(') 2 + -667c 2 ] 3/2 1 r J + r . (V.) In a recent article it was shown by Prof. W. L. Bragg * that the diameter of the neon atom could be found by an inspection of the diameters of the atoms of elements whose atomic numbers were near that of neon. It is impossible to measure the radius of the neon atom directly, since it forms no chemical compounds. The value obtained was very much smaller than that found by gas measurements, and the former is considered by Bragg to be the distance between the elec- trons in the atom — that is to say, is equal to ' 2c/ The value obtained by Chapman f from gas measurements is, however, the diameter presented by the molecule when in collision with other molecules. The difference is due to the fact that in molecular collisions in the gaseous state the outer electrons of the molecules do not come into contact}:. Using Bragg's value we have 2c = l'30 X 10~ 9 cm., and on substituting in equation (V.) we have a means of obtaining the fundamental value of l r.' An inspection of equations (II.) and (III.) shows at once that o) 1 = o) 2 and equations may be obtained for co 1 and co 3 . * Phil. Mag. vol. xl. p. 169. t Trans. Roy. Soc. A. vol. 216, p 279. X Rankine, Proc. Roy. Soc. vol. xcviii. p. 360. 208 Mr. L. St. 0. Broughall on Theoretical These equations take the form e 2 ( 39 , r r + l r — l ~]\ — 2 2mr 1 4r 2 L{F+F+^P + ■{ (r " 0' + s ' P /2 J J ~~ ^ ' .... (VI.) e 2 r f 1 11 ml { [( r -Z) 2 + 5 2 ] 3 / 2 " [(r + /) 2 + 5 2 ] 3 / 2 J = G, i 2 - ft) 3 2 . (VII.) Using a slightly different value for ' m ' from that used in the previous paper, we obtain the following values : — 6*19 x 10" 9 cm. 6-034 x 10 16 rad/sec. 4*290 x 10 16 rad/sec, 'm' being equal to 9*005 x 10 _28 grm. and £ = 4'774 x 10 -10 E.S.U. These figures refer to the neon atom when in its normal state. There is some doubt as to whether they apply without modification in the gaseous state, but certain assumptions are made later in this paper which leads one to the conclusion that if the atom is larger under natural conditions, then the only result will be the elimination of certain spectral lines in the ultra-violet. When the atoms of the neighbouring- elements were submitted to measurement, they constituted a solid body ; it is, therefore, quite conceivable that modi- fication will occur if the element becomes gaseous. In order to explain the nature of the spectral lines, we have to consider the change of energy due to a change of orbit, energy being emitted when the orbit increases in diameter. Bohr, as already stated, imagined in the case of hydrogen that the radius of the orbit increased by constant multiples of the radius of the initial orbit. To adopt such a plan in the case of neon would lead to the emission of spectral lines of a frequency which would only give ultra-violet lines under reasonable circumstances. Further, there is no reason why the increment should be of such a nature, and the hypothesis used in our case is that the spherical shell formed by the inner electrons increases in radius until the shell has a radius equal to that of the initial outer shell of electrons. In order that equilibrium may remain, it is essential that the outer shell also expands to an extent which can be calculated from equation (V.). The initial increment is of the nature of 3xl0~ 10 cm. This process of expansion continues again and again, the inner electrons always occupying the orbit previously occupied by the outer electrons. Aspects of the Neon Spectrum. 209 Having thus found a definite value for l r' corresponding to a definite value for * c' we are now in a position to calculate the values of the angular velocities about the ... several axes. To do this, equation (VI.) is first applied ; and having found ' g>jV the value obtained is substituted in equation (VII.), and &) 3 2 obtained. In calculating the value of ' c,' use is made of the fact that the ratio c : r is practically constant ; such ratio values are used in Table II. The closeness of the ratio figures and the absolute figures is shown in the appended table. Table I. 6-50xl0" 9 cm. 12-33 „ 1501 c (ratio method). c (from eqn. V.). 6-83xl0 _9 cm. 6-83xl0~ 9 cm. 12-95 „ 1295 15-77 „ 15-77 It will be seen that the agreement of the figures is so nearly exact as to warrant their use, remembering that our fundamental value of { c ; has not been obtained experi- mentally. Table II. shows in columns II. and III. the values of l c' and t r/ and in columns IV. and V. the values of w^ and ft) 3 - respectively. Table II. I II. III. IV. 1 No. Eadius Outer Orbit = c. Radius Inner Orbit = r. o h \ , 1 1 ... 6-50xl0 _9 cm. 6-19xl0" 9 cm. 36-41 X10 32 18-40 xlO 32 | 2 ... 6-83 6-50 3141 „ 15-84 3 ... 717 6-83 27-08 „ 13-65 „ 4 ... 7*54 „ 717 2336 „ 11-78 „ 5 ... 7-92 7-54 20-15 „ 10-16 „ 6 ... 8-32 7 92 17-38 „ 8-768 „ 7 ... 8-74 832 15-00 „ 7-568 „ 8 ... 9-18 8-74 12-93 „ 6-528 „ 9 ... 964 9-18 ' 1115 „ 5-620 „ 10 ... 1013 964 9-625 „ 4860 „ 11 ... 10-64 10-13 8-299 „ 4182 „ 12 ... 11-18 1064 7160 „ 3613 „ 13 ... 11-74 1118 6170 „ 3-109 „ 14 .. 1233 1174 5-326 „ 2-684 „ 15 ... 12-95 12-33 4-602 „ 2-322 „ 16 ... 13-60 12-95 . 3-970 „ 2-002 „ 17 ... 14-29 1360 3428 „ 1-729 „ 18 ... 15 01 14-29 2-954 „ 1-489 ,, 19 ... 15-77 1501 2-550 „ 1-286 „ 20 ... 16-57 1577 2-201 „ 1113 „ 21 - 17-41 1657 1-897 „ •9581 „ 1 Phil. Mag. S. 6. Vol. 44. No. 259. July 1922. 210 Mr. L. St. C. Broughall on Theoretical Let us now consider the energy of an electron in the outer shell when the diameter of the shell = 2^. Let the angular velocities about the axes YY' and ZZ' =w 1 . Let the angular velocity about axis XX / = W 3 . Since the diameter of the shell is equal to ' 2c 1? ' it follows that the radius of the electron orbit =Sj where Sx = C! v 6/3. Using the above notation and remembering farther that the kinetic energy of a particle describing a circular path of radius ' R ' with an angular velocity 'W is equal to ^MR 2 W 2 , where M is the mass of the particle, we find that the kinetic energy due to rotation about the axis XX' = ±mS{ 2 W and the kinetic energy due to rotation about the axis YY' plus that due to rotation about ZZ' = mS 1 2 W 1 2 since Wi = W 2 . Therefore the total kinetic energy of the particle is equal to >S 1 2 {2W 1 2 + W 3 2 }. In the case of an electron in the inner shell where the radius of the orbit =Rj, we have the kinetic energy of the electron due to its rotation about the axes YY' and ZZ'=E X where E 1 =wR 1 2 W 1 2 . Now let the inner shell expand until it occupies the space previously occupied by the outer shell — that is to say, until Ri = Ci, then kinetic energy in new orbit = E 2 » Where E 2 = ?nC 1 2 W 11 2 , 'W n ' being the new angular velocity about the axes YY' and ZZ', the change of energy, = E 1 -E» = m(R 1 2 W 1 2 -C 1 2 W 11 2 ). Meanwhile the outer electrons have moved further away from the nucleus, and now the outer shell has a radius = G 2 , and the orbit of the electrons is now S 2 . Therefore the energy in the new position is equal to *mS 1 *[2W 11 J + W 1 ,»]. Where ' Wi 3 ' is the new angular velocity about the axis Aspects of the Neon Spectrum. 21.1 XX', the change of energy due to change of orbit is therefore equal to im{S l 2 [2\V + W 3 2 ]-S 2 2 [2W„ 2 + W I3 2 ]}. Now, by Bohr's assumptions we have the equation Energy Emitted = Frequency x h. Now, the frequency of a light-wave =c/X where l c' is the velocity of light and i \ i is the wave-length. Therefore ch Energy Emitted = e= — , ch or X= — . e Now, in the case of an inner electron ch \= mCR^WV-L^VVn*)' giving a series of lines for different values of R. In the case of the outer electrons, 2ch X ~ m{S{\2 W * + W 3 *) + S 9 »(2 W n 2 + W„«) } ' giving a second series of spectral lines. Table III. shows the energies corresponding to definite radii. Column II. shows the energy content of an inner electron on the left, and that of an outer electron on the right. Column III. shows the change of energy, and column IV. shows the wave-lengths of the spectral lines produced. It should be stated here that the energy under consideration is the energy of one electron and not of I he whole shell. It has been stated that there is a possibility oE the atom not being in its normal condition to begin with, owing to its gaseous condition. If, however, it has expanded, then instead of starting with an atom whose diameter is 1'oOxlO -8 cm., we start with one whose radius is in all probability equal to one of the radii given in Table II. If this is so, then the only change produced will con- sist of the elimination of some of the lines of higher frequency. P 2 212 Mr. L. St. C. Brouffhall on Theoretical Table III. I. II a. It b, III a. Ill b. IV a. IV b. No. Energy per electron. Energy Difference (ergs). Wave-length of Spectral Line. 1 .. Inner. Outer. Inner. Outer. Inner. Outer. 1-255X10" 10 1-157 XlO- 10 2 .. 1-195 „ 1-101 „ 6-0xl0- ]2 5-6X10" 12 3272 A 3505 A° 3 .. 1-137 „ 1-048 „ 5-8 „ 5-3 „ 3386 „ 3704 ;, 4 .. 1-082 „ •9970 „ 5-5 „ 5-1 „ 3570 „ 3850 ,/ 5 .. 1030 „ •9491 „ 5'2 „ 4-79 „ 3776 „ 4100 „ 6 .. •9801 „ •9035 ,; 4-96 „ 4-56 „ 3959 „ 4305 „ 7 .. •9337 „ •8604 „ 4-67 „ 4-31 „ 4205 „ 4555 „ 8 .. •8886 „ •8187 „ 4-51 ., 4-17 „ 4353 „ 4709 „ 9 .. •8457 „ •7793 „ 4-29 „ 3-94 „ 4576 „ 4983 „ 10 .. •8055 „ •7423 ., 4-02 „ 3-70 „ 4884 „ 5307 ,.. 11 .. •7665 „ '7063 „ 390 „ 3-60 „ 5034 „ 5454 „ 12 .. •7297 „ •6723 „ 3-68 „ 3-40 „ 5335 „ • 5775 ,. 13 .. •6940 „ •6396 „ 3-57 „ 3-27 „ 5499 „ 6005 „ 14 .. •6609 „ •6089 „ 3-31 „ 3-07 „ 5932 „ 6396 ., 15 .. •6298 „ •5803 „ 311 „ 2-86 „ 6313 „ 6864 „ 16 .. •5995 „ •5525 „ 303 „ 2-78 „ 64S0 „ 7063 „ 17 .. •5709 „ •5260 „ 2-86 „ 265 „ 6864 „ 7409 „ 18 .. •5430 „ •5003 „ 2-79 „ 2-57 „ 7037 „ 7640 „ 19 .. •5172 „ •4766 ,, 2-58 „ 2-37 „ 7610 „ 8285 ,. 20 .. •4927 ., •4540 „ 2-45 „ 226 „ 8013 „ 8688 „ 21 .. •4689 „ •4321 „ 2-38 „ 2-19 „ 8249 „ 8966 „ Owing to the complexity of the neon spectrum, it would be useless to attempt to compare our calculated lines with those found by experiment ; indeed, it would be deleterious to attempt such a comparison, since the impression would be given that there is a definite line in the spectrum which corresponds to one of our calculated lines. Emphasis may only be laid upon the fact that our series produce lines in the visible part of the spectrum, which do not compare unfavourably with those obtained by experiment. Reference to Table III. will show at once that only forty lines have been determined between X = 3272 A and \ = 8966A, L Aspects of the JSfeon Spectrum. 213 whereas there are many more lines in existence. These lines can only be explained by the fact that when the spectrum is obtained, large numbers of ionized atoms exist, and under such circumstances our fundamental equations no longer hold. The discussion of the properties of ionized atoms is very complex, since the possible degrees and modes of ionization are very numerous. The first case which comes under con- sideration is the atom which has lost one electron, thus leaving an excess of one positive charge. It is very probable that one of the outer electrons will be removed, thus leaving seven electrons in the outer shell. Now, it seems probable that the angular momentums of the remaining electrons will suffer no change, the light pro- duced during ionization being due solely to the change of energy of the electron suffering removal. We are thus left with seven electrons, each possessing the same angular velocity. It is a matter of considerable difficulty to arrange these electrons, and it is impossible to arrange them on a spherical surface without the force acting on an electron varying with the electron taken. We are therefore obliged to separate them into different shells. Now, since the angular momentums of our seven outer electrons are the same, it follows that, if they are not on the the same spherical surface, then they must be in motion relative to one another. Under such conditions the positions of the electrons will vary with time. Owing to the complexity of such a case, it seems impossible to treat the case mathematically without more experimental evidence. There are further atoms present which have been ionized to a greater extent, thus losing several electrons. Similar difficulties are met with in the cases of atoms with five or six electrons in the outer shell as in the case of seven electrons. The cases in which four or six electrons have been removed are, however, considerably simpler, since the electrons may then be given positions on a spherical surface such that the force acting on an electron is not dependent on the electron taken. In general, the atoms ionized to so great an extent will be comparatively few. In all our cases of ionized atoms, it must be remembered that it is not only the normal atom that is ionized ; an atom may have given out several spectral lines before it becomes ionized. So a 214 Theoretical Aspects of the Neon Spectrum. large number of lines will be obtained depending upon the state of the atom when one or more electrons are removed from it. There is another form of ionization which is worthy of consideration. That is the case in which an electron has succeeded in penetrating the atom and reached the nucleus, thus temporarily reducing the positive charge and therefore giving a negative ion. The fundamental mathematical expressions for such a case are found by extending our formulae for the neutral atom for the case where the charge on the nucleus is ' n ' instead of ten. The angular velocities about the several axes will remain unchanged ; and so only two equations will be required to determine the new values assumed by ' c ' and ' r,' the radii of the outer and inner shells respectively. We have three available equations ; and expressing them in the notation previously used, we obtain : (4w-l> 2 Z e 2 eH_ e 2 (r + l) e 2 (r~l) 4c 3 " ~ U 2 + 2s* + [ (r +■ l) 2 + s 2 ] *' 2 ~~ [ (r - I) 2 + s 2 ] 3 / 2 + 2mo?l, .... (la.) (4n-l> 2 / e 2 eH e 2 l eH 4c 3 ~ - 4/ 2 + 2? + [(r + iy + s 2 y/ 2+ [{r-l) 2 + s 2 ] s < 2 -hmco 1 2 l-\-?nco :i 2 l, . . (II a.) ne* <ke 2 (r + l) 4<? 2 (r-Q + Vt*. 7\2 i .213/2 + + 2mft> 1 V, . . . .(Ilia.) which are obtained from equations (I.), (II.), and (IV.), replacing the nucleic charge of ' 10<? ' by ' ne 9 and remem- bering that &) 1 = G) 2 . The result of the alteration will be that the electrons will move further out from the nucleus, since i n } is of necessity less than ten. In consequence of this, the frequency of the spectral lines produced by such ionized atoms will be of a lower frequency than those produced by the neutral atom. There will in consequence be a larger number of lines in the part of the spectrum of greater wave-length. It is for this reason that there are so many lines in the orange, and red in the case of neon. Feb. 13, 1922. 1. [ 215 ] XV. Absorption of .Hydrogen by Elements in the Electric Discharge Tube. By F. H. Newman, Ph.D., F. Inst. P., A.P.C.S., Head of the Physics Department, University College, Exeter *. 1. Introduction. rjpHE phenomenon of the disappearance of gas in the JL electric discharge-tube, and in the presence o£ incan- descent filaments, has received much attention recently owing to its importance in technical applications. Langmuir f has shown that hydrogen disappears from a vacuum tube in which a tungsten filament is heated above 1000° C. # This fact has been utilized by him in the removal of the last traces of gas in valves, and the effect has been termed a " cleaning up " one. The pressures at which he w r orked were very low ; for example, he found that the pressure in a tube was lowered to 0*00002 mm. of Hg. Other gases, including nitrogen and carbon monoxide, are removed in a similar manner, and molybdenum, when incandescent, has the same effect as tungsten. In all cases Langmuir found that the cooling of part of the apparatus by means of an enclosure at liquid- air temperature greatly accelerated the rate of disappearance of the gases. In addition he noted an electro-chemical " clean up," which occurred at much lower temperatures of the filament, when potentials of over 40 volts were used in a way that caused a perceptible discharge through the gas. More recently Campbell, conducting work for the General Electric Company J and using incandescent filament cathodes in electric discharge-tubes, has made an exhaustive study of the " clean up " effect, and has come to the conclusion that there is much evidence for believing there exists an electrical action which is quite independent of the thermal action, and, providing the temperature of the filament is kept below that at which the chemical " clean up " occurs, the effect appears to be one dependent only on the electrical discharge. In the case of the disappearance of carbon monoxide there is proof of the conversion of this gas into carbon dioxide, and the action takes place more rapidly when part of the apparatus is cooled to liquid-air temperature. This has the effect of removing the carbon dioxide by condensation as quickly * Communicated by the Author. t Am. Chem. Soc. Journ. vol. xxxvii. (1915). X Phil. Mag. vol. xl. (1920) ; vol. xli. (1921) & vol. xlii. (1921). 216 Dr. F. H. Newman on Absorption of Hydrogen as it is formed. The presence of phosphorus vapoui accelerates the rate of disappearance of all gases except the inert ones, and much lower final pressures are attained. This, the author believes, is due to the deposition of the gas on the walls of the vessel, this deposit then being covered with a layer of red phosphorus formed by the electric discharge passing through the phosphorus vapour. The covering of red phosphorus prevents liberation of the hydrogen by bombardment of the ions, and at the same time provides a new surface on which further gas can be deposited. The problem of the disappearance of the gas is a very complicated one, owing to the many factors to be considered. The walls of the vessel and the electrodes will certainly receive some of the gas, although the latter may not disappear in its original state. There will be chemical changes occurring in the volume of the gas, such as the conversion of carbon monoxide into carbon dioxide, and, in addition, any other elements present in the discharge-tube, either in the form of vapour or solid, will affect materially the rate of dis- appearance of the gas and the final pressure reached. The author * has shown previously that various substances present on the electrodes of a discharge-tube alter considerably the amount of gas that can be caused to disappear when an electric discharge is passing. In particular, phosphorus, sulphur, and iodine cause both hydrogen and nitrogen to be absorbed at a very great rate, and a high vacuum is quickly produced as a result. This action of phosphorus has been used for many years to obtain and maintain very low pressures in valves. These three elements stand out as being- very effective even at high pressures, but other substances which were tested in a similar manner did not appear to absorb hydrogen. On the contrary, gas appeared to be liberated. This effect can be explained as follows. At pressures above 1 mm. of Hg. a certain amount of the gas in a discharge-tube becomes occluded within the walls. This gas will be liberated when the walls are bombarded by the ions produced by an electric discharge. This effect will mask any disappearance. If, however, the tube is heated almost to the softening point of glass and highly exhausted, then on admitting hydrogen at a small pressure such as O'l mm. of Hg., very little occlusion of the gas within the walls will take place, and on passing the electric * Newman, Proc. Hoy. Soc, A. vol. xc. (1914); Proc. Ph. vs. Soc. vol. xxxii. (1920) & vol. xxxiii. (1921). by Elements in the Electric Discharge- Tube. 217 discharge practically no hydrogen will be liberated from the walls by bombardment with the ions. If there is any absorption of the gas, this effect will not be masked by the liberation of the gas from the walls or electrodes. The object of the present work was the study of the behaviour of hydrogen in the presence of various elements in a discharge-tube when a current was passing through it. The pressures of the gas in these experiments were much lower than those used by the author in the papers quoted above, but they were much greater than those used by the previous investigators — Langmuir and the General Electric Campany. 2. Description of Apparatus. At gas-pressures below O'l mni.Hg.it is difficult to obtain a current through a discharge-tube unless very high potentials are used. A valve also must be placed in the circuit to make the discharge unidirectional. This entails further diminution Fiff. 1. M c LEOD GAUGE 3=fcdl^b- of the current. By using a Wehnelt cathode the potential required was greatly reduced. The apparatus employed is shown in fig. 1. The incandescent filament was a strip of platinum foil 5 mm. long and 3 mm. wide. As the discharge-tube had to be thoroughly cleaned after each experiment, the cathode was sealed in a glass stopper which could he removed when the tube was cleaned. This necessitated the use of tap-grease, but the vapour arising from it did not appear to afreet the results at the pressures used. 218 Dr. F. H. Newman on Absorption of Hydrogen Previous experiments had shown that elements such as sodium and potassium are only effective in causing the disappearance of gas in the electric discharge-tube if the surface of the element is clean, and if it has been prepared in vacuum. Accordingly the substance under test was placed on the platinum foil forming the cathode, and after the tube had been heated almost to the softening point of: glass and exhausted, the element on the foil was vaporized by passing an electric current through the latter. In this way the substance was then deposited on the inner surface of the anode D and an uncontaminated surface obtained. The anode was of aluminium and was cylindrical in shape, fitting very closely to the glass walls. Enclosing the cathode in this way, the effect of the surface of the glass on the absorption of the gas was minimized. A side tube B was used to contain the easily volatile elements such as phosphorus, sulphur, and iodine. An aperture was made in the anode opposite the mouth of B so that the vapour ot the substance from B could pass through and be deposited on the inner surface of D. The pressures of the gas were measured with a McLeod gauge. The hydrogen was prepared by the electrolysis of barium hydroxide and stored in a reservoir. This method of preparing the gas ensues great purity. Any oxygen present was removed by passing the gas through a bulb containing sodium-potassium alloy. Phosphorus pent- oxide in F removed any water-vapour, and of course the alloy was effective in this respect also. The gas could be admitted to G, which was a known volume (0*051 c.c.) enclosed between two taps. A definite volume of gas at a known pressure could thus be admitted to the discharge-tube. From obser- vations of the pressure in the tube before and after a discharge had passed, the actual volume of gas — at atmospheric pressure — which had disappeared could be calculated. The current through the discharge-tube was kept constant by altering the filament current, and was measured with a galvanometer. In previous experiments the quantity of electricity passing through the tube while absorption was taking place had been measured with a water voltameter, but in the present work this method was not sensitive enough. After deposition of the substance on the anode D, the tube was again highly exhausted to remove any gases liberated from the volatized substance. The tube was placed in an enclosure maintained at —40° C. while absorption of gas was in progress. by Elements in the Electric Discharge- Tube. 219 Observations were then taken of the changes in pressure due to the disappearance o£ the hydrogen when an electric discharge passed through the gas. The results obtained are shown in the accompanying table. The accelerating potential was 94 volts, obtained by using small accumulators. The current through the tube was kept constant, and was 546 micro-amps. Each set of readings corresponds to an electric discharge for ten minutes, except in the cases of sulphur, phosphorus, and iodine, where the observations were taken at intervals of two minutes — i. <?., with sodium the pressure changed from 743 xlO" 3 mm. of Hg. to 336 x 10" 3 mm. of Hg. in ten minutes, while with sulphur the pressure was lowered from 740 xlO" 3 mm. of Hg. to 329 x lO" 3 mm. of Hg. in two minutes. The amount of hydrogen which would be liberated from a water voltameter in 10 mins. by the same current is 39 x 10~ 3 c.c. at atmospheric pressure. As the gas may disappear into the walls of the anode even in the absence of any substance on the anode, and as the glowing filament may affect the rate of disappearance, preliminary observations were always made when an electric discharge passed through the tube without the substance present on the anode. The volume of gas which disappeared owing to these two effects was always very small compared with that which was absorbed when the element under test was on the anode. 3. Experimental Results. After each element had been tested, the tube was heated to 300° C, and the volume of gas reliberated was calculated from the observed change of pressure. The amount thus recovered varied considerably, but was always less than that which had disappeared. This evolved gas was again absorbed when a discharge was passed, and it is evidently hydrogen in the same condition as it was before disappearance. If, after the gas had disappeared, a fresh amount of hydrogen was admitted, the volume which disappeared on discharge was reduced. For example, with sodium and the gas pressure at 743 X 10" 3 mm. of Hg., the vacuum was reduced to 96xl0" 3 mm. Hg. before the action ceased. Admitting a further supply of gas to the tube, the pressure fell from 743 xlO" 3 mm. Hg. to 233 xlO" 3 mm. Hg. Repeating the process again, the pressure fell from 743 xlO" 3 mm. Hg. to 436 x 10" 3 mm. Hg., and then the 220 Dr. F. H. Newman on Absorption of Hydrogen Element. Initial Gas Pressure. mm. H?. X 10-3 Final Gas Pressure. mm.Hg.XlO-3. Volume of Gas absorbed. c.c. x 10-3. Pressure of Gas at which action ceased. mm.Hg.XlO -3 . Sodium Potassium Sodium-Potassium Alloy. T743 1336 726 349 738 392 336 123 349 163 392 188 329 145 58 f740 Sulphur i 329 f748 Phosphorus -{352 1200 t j- ' 763 Iodme 1338 Arsenic {394 Cadmium j ^o Calcium |^ 04 »»■': {23 ry J 753 Zmc J431 Thallium {560 j , j\ Hydrogen was liberated and not absorbed 352 147 44 338 150 394 206 463 321 404 386 264 431 306 560 399 28 14 26 13 23 15 28 13 6 27 14 7 28 13 24 13 19 10 23 7 25 8 22 8 12 11 96 84 110 26 14 124 108 284 152 131 276 297 absorption ceased. There appears to be a fatigue effect whereby the actual amount of gas which can be absorbed by any surface is limited. This fatigue effect may be due to three causes. If the disappearance of the gas depends on chemical action, the latter will occur mainly at the surface of the element. The formation of a chemical compound will thus protect the rest of the substance from the action, and the process will gradually cease. If, on the other hand, the effect is due to a deposition of the gaseous atoms on by Elements in the Electric Discharge- Tube. 221 the surface, as Langmuir suggests, these atoms will diffuse slowly into the substance. The atoms arriving later will have less area <>n which deposition can take place. A limit to the action will be reached when the number of atoms deposited is equal to the number set free by the bombard- ment of the surface by the ions. After absorption, the proportion of the hydrogen re- liberated when the tube was heated to 300° 0. varied considerably in different cases, not only with different elements, bat also with the same element. This is to be expected when it is remembered that the thickness of the substance deposited on the anode varied with different substances. The accelerating potential affected to some extent the rate of disappearance of the gas and also the final pressure attained. Owing to liberation of the gas by the bombardment with the ions, the final pressure reached must depend on this reverse action, and the greater the accumulation of the gas on the surface of the anode, the greater will be the amount of gas evolved. With sodium on the anode a potential of 94 volts reduced the gas-pressure from 743 X 10~ 3 mm. Hg. to 123 x 10 _3 mm. Hg. in the course of 20 minutes. When the potential was lowered to 54 volts, the pressure fell from 743 x 10 ~ 3 mm. Hg. to 476 x 10" 3 mm. Hg. in the same time-interval. The final pressures reached before absorption ceased were 96xl0" 3 mm. Hg. and 202 xlO" 3 mm. Hg. respectively. The current through the discharge-tube was kept constant throughout. The principles of the disappearance of the gas will be discussed later, but there are certain features of the phenomenon which can be traced to chemical actions. Many of the elements tested combine with hydrogen at high temperatures to form chemical compounds which are very stable. Any chemical action occurring in the present experiments cannot be due to the heat, as the discharge-tube was maintained at —40° C, and the incandescent filament was always at a lower temperature than that at which Langmuir found chemical action occurred with hydrogen. The effect may be caused by " activation " of the gas, the latter assuming some modification under the action of the electric discharge. In the above experiments the amounts of gas absorbed were so small that it would be extremely difficult to detect the existence of any chemical compound in the tube. In order to increase the amount of gas absorbed, 222 Dr. F. H. Newman on Absorption of Hydrogen and test for any chemical compounds formed, a modified form of the apparatus was used, as shown in fig. 2. Pure hydrogen could b.j admitted to the discharge-tube A in small volumes by manipulation of the taps 1\ and T 2 . Two strips of platinum foil, about 10 cms. long, were sealed in the tube E. These strips fitted closely to the glass surface. A potential difference of 600 volts was applied between these strips by means of small accumulators. In this way the ions actually present in E were removed while the discharge was proceeding in A. The tube E communi- cated with a mercury cut-off K, and a U tube immersed in an enclosure maintained at —40° C. Sodium-potassium alloy was prepared in D, and after the whole of the apparatus had been evacuated, the alloy was run into C. In this way a bright and clean* surface was obtained on the alloy in C. A n \ HYDROGEN small volume of hydrogen was then admitted to A, the mercury cut-off preventing the gas from entering C and D. While the electric discharge was passing in A, the hydrogen was allowed to enter C by manipulation of the cut-off. Admitting successive volumes of hydrogen into A in this way, and each time allowing communication with C while the discharge was proceeding, an increasing amount of active gas entered C, and an effect was observed on the surface of the alloy. At first it appeared to be covered with a thin white crystalline compound when observed through a microscope. This white layer slowly changed, on the admission of more active gas, to a dark grey-coloured deposit. To show that this surface effect was not due to impurities in the hydrogen, previous experiments were made, the gas being admitted to C without the electric discharge proceeding. There were no surface effects then, by Elements in the Electric Discharge- lube'. 223 so it was concluded that some of the hydrogen assumes an active modification under the action of the electric discharge, and in this form it is able to form chemical compounds with the sodium and the potassium present in the alloy. The U tube in the enclosure at —40° C. excluded the possibility of the action being due to the heat from the discharge-tube. The white crystalline compound which first appears is a mixture of the hydrides of sodium and potassium. The exact nature of the greyish-coloured product formed afterwards is unknown, but it is probably a solution of the hydrides in the alloy. Water is evolved by an electric discharge when passed through any vacuum vessel. It comes from the glass, and would not be kept back by the trap cooled to —40° C; for at that temperature water substance has a vapour-pressure of about 0*1 mm. Hg. The presence of water-vapour " fouls " the surface of the alloy, but this fouling gives a black deposit on the surface which is quite different from the white crystalline layer observed in the present experiments. The black deposit consists of sub-oxides of sodium and potassium, and its appearance has been noted previously by the author *, although in rthat paper it was attributed to the hydrides. It has now been proved by chemical analysis that this black deposit does consist of the sub-oxides. Sulphur was tested in the following manner: — A small piece of filter-paper, soaked in lead-acetate solution, was placed together with a small amount of the solution in D. The rest of the apparatus was separated from D by a mercury cut-off not shown in the figure. C contained sulphur which had been deposited in a thin film over the interior. After exhausting the whole of the apparatus to a pressure of about 5 mm. of Hg., the mercury cut-off between C and D was closed and the rest of the apparatus highly exhausted. Hydrogen was then admitted to A until the pressure was about 7 mm. Hg. While the electric discharge was passing, the mercury cut-off was opened. This was repeated many times, the pressure of the gas in A being gradually increased. Each time communication with D was established, any gaseous product formed in C was admitted to D. In the course of a few minutes the paper soaked with the lead- acetate solution turned black, showing the presence of a sulphide of hydrogen. This chemical compound must have been produced by the action of an active form of hydrogen on the sulphur. The surface of the mercury at the cut-off * Proc. Roy. Soc. A. vol. xc. (1914). 224: Dr. F. H. Newman on Absorption of Hydrogen also lost its bright appearance. This was due to the action on it o£ the sulphide of hydrogen. The mercury surface remained quite clear when the sodium-potassium alloy was tested. Sulphur and the alloy were selected for tests because the chemical actions in these cases give rise to compounds whose effects can be noted easily. It is extremely difficult to examine phosphorus and iodine in this way owing to their high vapour-pressures. A possible test would be the com- parison of the vapour-pressures before and after absorption of hydrogen had taken place. These two experiments indicate that the chemical action is not due directly to the ions in the discharge-tube, as they were all eliminated by the charged platinum strips before reaching either the alloy or the sulphur. Wendt * showed that hydrogen can be activated by the passage of a-rays through the gas, and it has been shown by the author f that the active modification so produced is able to react chemically with sulphur and the alloy of sodium and potassium. 4. Discussion of Results. The disappearance of gas in a vacuum-tube is due probably to several principles, some of which may be fundamental. It is certain, however, that any attempt to explain the principles by the same theory would lead to conflicting- results, but the processes occurring can be divided into two classes, chemical and mechanical. There is much evidence that the gas can be caused by the electric discharge to adhere to the solid parts of the discharge-tube in some manner which is at present unknown. In many cases a portion of the gas can be reliberated by heating the vessel, but no reason can be advanced for the non- reliberation of the whole of the gas which has disappeared. Langmuir assumes in the paper previously quoted that the hydrogen in the presence of an incandescent filament under- goes dissociation. The gas shows abnormal thermal conductivity at high temperatures, due to its atomic nature. The dissociation does not occur apparently in the space round the wire, and is not due to the impacts of the gas molecules against its surface, but takes place only among the hydrogen molecules which have been absorbed by the metal of the wire. Some of the atoms leaving the wire do not meet other atoms, * Nat. Acad. Sci. Proc. vol. v. (1919). t Phil. Mag. vol. xliii. (1922). by Elements in the Electric Discharge- Tube, 225 owing to the low pressure, but diffuse into the tube cooled by liquid air, or become absorbed by the glass, and thus remain in the atomic condition. They retain all the chemical activity of the atoms. Langmuir also found that when the liquid air was removed, some of the atoms would come off the glass and recombine with other atoms to form molecules. These molecules could not be recondensed by replacing the liquid air. This gas which would not again disappear he termed a " non-recondensible" gas. This hypothesis, which is applicable to very low pressures, cannot hold at the pressures used in the present work. The gas in the atomic condition can scarcely move from the discharge-tube for a considerable distance and still retain its atomic nature. The " non-recondensible " gas found by Langmuir is probably hydrogen in its normal state. When nitrogen gas disappears in the discharge-tube, practically none of it can be reliberated, even when the tube is heated to the softening point. This fact indicates a striking difference between the disappearance of hydrogen and nitrogen. If chemical compounds are formed by the absorption of the gases, this difference can be explained in terms of the difference in the stability of the hydrides and nitrides produced. The chemical action may take place between hydrogen and the vapour of the element, and also it may occur at the surface of the solid. The majority of the elements studied have such small vapour-pressures that a very small propor- tion of the action is due to the vapour. The active condition of the gas must be caused by the ions, although results seem to indicate that the number of active atoms or molecules in the gas is of a much higher order than the number of ions present in the gas at the instant of recombination. The absorption is not due entirely to chemical action, as the law of constant proportions does not seem to be followed. It is of significance, however, that the rate of disappear- ance of the gas increases, and the final pressure attained decreases, as the temperature of the discharge vessel is lowered. This arises from the lowering of the vapour- pressure of the compounds produced, with the result that the final pressure reached is lowered. Although the formation of hydrogen sulphide in the discharge-tube by the action of the activated hydrogen on the sulphur will not explain the disappearance of the gas, it does indicate the production of a modified form of the gas which is able, possibly, to form other compounds with sulphur in addition to hydrogen sulphide. Phil Mag. S. 6. Vol 44. No. 259. July 1922. Q 226 Mr. Bernard Cavanagh on That the mechanical deposition o£ gas on the walls of the discharge vessel will not account entirely for the disappear- ance of the gas is shown by the difference in the behaviour of nitrogen and hydrogen with phosphorus, sulphur, and iodine. Practically none of the nitrogen can be reliberated by heating, but a large proportion of the hydrogen is evolved. There is reason for believing that the modification of hydrogen is triatomic in nature. Wendt has shown in the paper previously quoted that hydrogen drawn from a tube through which an electric discharge is passing contains a small quantity of H 3 . Probably monatomic hydrogen is first formed, and owing to collision with neutral molecules of the gas, H 3 then appears. The monatomic gas may be produced originally by the action of the swift-moving electrons on the molecules. Wendt and Grubb * have also shown that N 3 is produced when an electric discharge passes through nitrogen. Thomson f found evidence of H 3 in his positive-ray experi- ments. It is this triatomic form of hydrogen which is effective in the production of chemical compounds in the electric discharge-tube. XVI. Molecular Thermodynamics. II. By Bernard A. M. Cavanagh, B.A., Balliol College, Oxford %. I. Molecules, Thermodynamics, and Quantum Theory. IN developing a molecular treatment of the thermodynamics of dilute solutions in simple solvents, Planck § deter- mined the form of the integration constants in the entropy function by a method which was at the time the subject of some controversy. M. Cantor || objected that the hypothetical transition to the gaseous state without change of the molecular composition was not even theoretically possible, since there probably existed in the liquid state, complex molecules whose existence was inseparably connected with the condensed state of the phase, and entirely incompatible with a state of high temperature and low pressure. * Science, vol. lii. (1920). t Proc. Roy. Soc. A. vol. Ixxxix. (1913). t Communicated by Dr. J. W. Nicholson, F.R.S. § « Thermodynamics,' 1917 (Trans. Ogg), pp. 225-226. Or see Phil. Mag. xliii. p. 608 (1922). 'I Ann. der Phys. x. p. 205 (1903). Molecular Thermodynamics. 227 In reply, Planck * pointed out that the theoretical pos- sibility of the ideal transition depended only on the fact thai the numbers of the various molecular species were, together with temperature and pressure, the independent variables which determined the phase. Now the present author would suggest that Planck's reply can be construed (and, to be unanswerable, must be construed) as a rider to the definition of the terms " molecule " and " chemical compound," for the purposes of molecular thermodynamics, viz. : — " That the numbers of the various molecular species can be considered, together ivith temperature and pressure, as the independent variables determining the phase/'' or, in other words, " that it shall be theoretically sound to conceive any desired change of temperature and pressure of the system as taking place without change in the numbers present of the various molecular species f.." Oniy with this rider to our definition can it be laid down, for instance, that the "mass-action" equilibrium law must be obeyed in sufficiently dilute solution, for it is to be observed that purely " general " thermodynamics has no cognisance of molecules, but takes for its independent variables, besides temperature and pressure, the masses of the "components.'" [See next section of this paper.] The misconceptions which have so long stood in the way of a satisfactory general theory of electrolytic or "con- ducting " solutions seem sufficiently to illustrate the indis- pensability of this postulate. A parallel illustrating its signih'cance may be drawn from the dynamical theory of chemical combination and dis- sociation. The classical dynamical conception of a binary molecule (for example) was a pnir of simpler molecules (or atoms) moving relatively to one another in closed orbits, and the principle of the conservation of energy forbade the spon- taneous dissociation of such a " molecule," requiring that its disruption should depend on collision with another molecule. The well-known fact that dissociation is (at constant temperature) independent of collision-frequency, showed the inadequacy of this conception, and pointed to some property, in the "forces" producing and maintaining a molecule, altogether incompatible with the older or "continuous" dynamics. * Ann. der Phys. x. p. 436 (1903).^ f Intermediate and tinal states being unstable, of course, in general. Q2 228 Mr. Bernard Oavanagh on The same difficulty arose when a dynamical explanation of the law of mass-action was attempted, the essential continuity of action of u physical " forces * standing in the way, and Boltzmann had to assume — with conscious artifici;tlity — dis- continuity in a field of force in order to arrive at the desired result. It seems indeed that, besides the more obvious and less peculiar properties of shortness of range, " specificity," and saturability, there is a quality of discontinuity of action (in time or in space or in both) which distinguishes "chemical forces '• from " physical," the distinction being sharp so far as we can yet see. We find then an absence of direct dependence of dissociation upon the thermal motion closely connected with the possibility of accounting for the law of mass-action dynamically. The parallel with our " rider " and its indispensability as a basis for the deductions of molecular thermodynamics is significant. The transitory orbital system which was the older " physical " conception of a molecule, and which quite probably occurs in all dense gases and liquids f, is clearly quite directly dependent on the thermal motion, is, in fact, itself merely an u episode " in that motion, and cannot in any sense be regarded as fulfilling the requirement of out- rider. We cannot, therefore, predict from thermodynamics the " mass-action " equilibrium law for the " reactions " of such " molecules " under any circumstances, for we cannot treat them 'as molecules for the purposes of molecular thermodynamics. And, in parallel, we find that dynamical theory is unable to predict the law of mass-action for such " molecules." The electrolyte question provides an important application of these considerations. It has frequently been supposed that a pair of ions, closely linked by their electrostatic fields alone, must be regarded as a molecule, and should behave thermodynamically as such. Electrostatic forces as we know them, however, are typical * That is, " forces " within the conception of the older physics. It seems convenient to use, in contradistinction, the term u chemical forces " for the " forces " or means by which a molecule is formed or /' hound " [see end of this section] and held together, and the expression may find some further justification in the fact that in the nature of these latter '' forces " lies, probably, the key to all the facts and phenomena of chemistry. t Compare here Cantor's objection, mentioned above. Molecular Thermodynamics, 229 " physical " forces, and it is our present conclusion that such forces are capable of forming only transitory associations — ' f episodes in the thermal motion " — essentially different from what we regard as molecules. \\ hen, as in the weaker acids and bases almost certainly, we really have partial ionization, or rather partial association of the ions to form u undissociated molecules," the latter must be regarded as produced and held together not by such ordinary electrostatic forces, but by " chemical forces " with the peculiar property already discussed. With regard to " strong electrolytes/' the work of Debye and Bragg oives good reason to believe that the molecule in the salt crystal is the ion. If this be so, it appears necessary to admit that the solid salt is essentially a mixed crystal whose special simplicity and homogeneity is due simply to the polarity of the electrostatic forces which dominate its " growth." Now, it would seem altogether inconsistent to suppose that the chemical " association " which does not take place in the intimate contact of the solid state, ensues when the ions are dispersed in a solvent, so that until the calculated effect of the electrostatic forces between the ions upon their thermo- dynamic behaviour can be shown to be inadequate when compared with experiment, the " complete-ionization " theory seems the only rational theory for strong electrolytes. Another application of the above general conclusions is to be found in the important question of " solvation " of solutes, which is treated in a paper to follow this. There occur in the literature of this subject such state- ments or suggestions as that " the solvates need not be definite chemical compounds," and vague theories of the " solvate molecule " as a mere indefinite conglomerate. From the preceding, at any rate, it is our conclusion that, unless the u solvate molecule " is produced and maintained by ' f chemical forces " in the sense already considered, so that it fulfils the requirement of our " rider," it will not, for the purposes of molecular thermodynamics, be a molecule at all. With regard to this remarkable characteristic of u chemical forces " which appears to be reduced to its lowest terms in the expression " discontinuity of action," this seems to mark out the problem of molecule formation (including, be it noted, reaction-velocity) as one of those many whose solution may be hoped for from the new quantum- dynamics of phenomena on the atomic scale. Indeed, it is tempting to believe that Bohr's conception of u electron-binding " may be the solution 230 Mr. Bernard Cavanagh on in embryo o£ the larger and more complex problem o£ " atom- binding/'' and that in his distinction between "bound " and " unbound " electrons in the atom, we may have in its simplest aspect the distinction between "chemical" and " physical " forces. II. Molecular Thermodynamics. In general thermodynamics, which is based on, and applies to experience, the independent variables, besides temperature and pressure, are the masses of the "components/' and these are reduced to the minimum necessary to define the system under all circumstances not conventionally or prac- tically excluded from consideration. To take a familiar example, hydrogen and oxygen will suffice as the components of a system containing in addition water, provided low temperatures are excluded from consideration, or the presence of efficient catalysts is assumed. In so far as we may suppose that the decomposition and formation of water do proceed even at low temperatures in the absence of catalysts, though at an immeasurably small rate, it is clear that theoretically the two components would always suffice for this system if sufficient time ivere allowed. And conversely they would never suffice if the rate of experi- mentation were sufficiently increased. Striking examples of the practical reality of this entry of the time factor into the question of the necessary number of components, have been given in recent years by the work of A. Smith and of A. Smits, who by increasing the rate of experiment ation, have increased the number of components necessary to describe certain systems, the latter author having propounded an interesting theory of allotropy on the basis of his experiments. Now we can conceive this carried far beyond the bounds of purely practical limitations, and the question arises, " How far ? * The atomic or elementary theory of matter is introduced when we say that at one extreme, when unlimited time is available, the elementary atomic species will be necessary as well as sufficient as the components of any system. Starting from this extreme and increasing the rate of experimentation we can imagine one complex after another of these elementary atoms (as its rate of formation and decomposition ceases to be great in comparison with the rate of experimentation) taking its place in the list of " com- ponents necessary to describe the system." Remembering, Molecular Thermodynamics. 231 however, that the masses of the components are together with temperature and pressure the independent variables, we see that there may be a limit, for it must be theoretically possible to alter temperature and pressure so quickly that the numbers present of these complexes which we are admitting as components remain sensibly unaltered during the change. According to the view put forward in the previous section, it involves us in a definite postulate bearing upon the nature of molecules and of chemical change, when we say that the limit is reached ichen and only ivhen every molecular species which can be formed in the system has taken its place in the list of ' ; components.''' Proceeding in theory to this limit, we obtain the general expression for -fy which is referred to in the sequel as the ".molecular expression for ty" and as Planck showed, we can determine it completely when it is linear by " connecting-up " with the known properties of the low-pressure gaseous mixture. When the expression is not linear, the higher or " general " terms are subject only to a single limitation inherent in Planck's method, as pointed out in the previous paper *. Observing that the " general " terms in the corresponding expression for U f do not involve " chemical" energy, the present author also suggested and illustrated J the interesting- possibility of employing ordinary dynamical theory, at least as a valuable aid, in determining and interpreting the form of these " general " terms. This "molecular expression for yfr," however, will clearly not in general correspond with our experiments carried out under ordinary conditions. They will correspond with an expression of the "general thermodynamic" type in which the components are appropriate to the conditions of ex- periment. The theoretical problem then presents itself of connecting this " experimental" expression for -\jr, in a manner at once rigorous — that is trustworthy — and practically effective, with the "molecular" expression and its possibilities of theoretical interpretation. The treatment of two important problems of this kind has been attempted. Planck pointed out that when a single molecular species * Phil. Mag. xliii. p. 606 (1922). t Entirely analogous considerations apply to V, but ordinarily owing to the low pressures used, V figures relatively negligibly in the determi- nation of ^ [c/. footnote, p. 630, Phil. Mag. xliii. (1922)]. X Loc. cit. p. 62"). 232 Mr Bernard Cavanagh on greatly preponderates, it is a matter of mathematical necessity that the " molecular expression for \jr " should take (in the limit) a linear form, and it was to this type o£ " dilute solution " that Planck confined himself, arriving readily at the Raoult-van't Hoff " laws of dilute solution." Dilute solutions in a liquid paraffin would be of this type. Van Laar * took the linear expression as the criterion of " perfect solution 5 ' in general, and not making the approxi- mations which Planck, considering very low concentrations, had made, was able to show that the Raoult-van't Hoff laws formed too restricted a criterion when the solution was very dilute. He, however, considered only solutions in which the solvent was of the same type as that of Planck, viz. : a single molecular species. In view of the fact that the Raoult-van't Hoff laws have been found to hold for dilute solution in our common and useful solvents, which are certainly not of the type considered by Planck and Van Laar, the present author was led to the problem of " complex solvents," which will be the first illus- tration of the theoretical problem outlined above. A preliminary treatment appeared in the first of these papers, but a more complete and rigorous treatment is now presented. The second illustration will be the problem of partially " solvated " solutes, a discussion of which will follow that of " complex solvents." The first result is that the Raoult-van't Hoff laws have been rigorously predicted for extreme dilutions in such solutions. It is shown, in fact, how the " experimental " expression for i/r simulates, in the limit, the " molecular " expression in form. But farther the way is prepared for the thorough investi- gation of middle and high concentrations in such solutions. To this end the "linear" terms in the experimental expression for i|r, which simulate and replace the simple linear terms in the molecular expression, have been treated with some thoroughness and rigour, these being the terms to which the expression reduces when the solution is " perfect." When the quite practical criteria thus provided are applied, the belief that " perfect solution " always ceases in these "complex" solutions at quite low concentrations may be largely dispelled. In simple solutions of the kind considered by Planck and * Z.f. Phys. Chem., several papers, 1903 etc. Molecular Thermodynamics. 233 Van Laar we find "perfect solution" persisting up to very high concentrations — sometimes over the whole range. The "general" terms of which (excepting the case of electrolytes) little or nothing is yet known, have been touched on only in so far as the treatment of the " linear " terms involves (in general) a certain very slight alteration in the division into u linear " and "general" terms, which may sometimes have to be taken into account when dynamical theory is employed. Experimental determination of the " general " terms for comparison with theory must of course be preceded by knowledge of the " linear " terms, — hence again the need for rigorous and thorough treatment of the latter. III. Complex Solvents. The importance of this question of " complex" (polymer- ized and mixed *) solvents is sufficiently obvious when it is considered that of this type are most of our best solvents, probably all our " ionizing " solvents, and, chief of all, water. We have to consider a solution consisting of the solute- molecular species, m l3 m 2 , , m s , . . . . , in addition to the various species m 01 , m 02 , . . . . , which constitute the solvent. Concentrations being expressed in gram-molecules per gram of solvent, the " molecular " expression for yjr is + S» s (&-Rlog^-J^-) +-RM S4, x 'f x (e 01 ... ,....), (1) which is of the form *-*"%+HWj- ■■ ■ > 2 > the several solvent-molecular species appearing as separate * Mixed solvents, while submitting to the same theoretical treatment as the merely polymerized solvents, present certain peculiar difficulties and some interesting possibilities with which it is hoped to deal at length in a later paper. t A suffix outside a bracket is used to indicate, in less obvious cases, independent variables which are held constant in a partial differentiation, — a well-known usage. A single suffix may be used briefly for a whole series, as n 0] here standing for w i, n 2, 234 Mr. Bernard Cavanaoh on components, whereas the " experimental '"' expression must have the form t = M„|J + 2 ras (|t) ..... ( 3 ) the solvent appearing as one component only. Clearly (*±\ = (^±) + f S ^r_ .3^oi] , . (4) \'dn s l m \dn,/n 0l L drc 01 ' dn s jM ' but the relation we have to use in making the change of variables is that given by the condition for chemical equi- librium among the molecules of the solvent, viz. : — rs|* .dHoii =o, (5) L On 01 J Mo so that \dn s J Mo \bn s / noi (6) and comparing then (2) and (3), BM;" Scol d^' (7) which could be regarded as physically obvious, as was done in the preliminary treatment (previous paper). We shall abbreviate (1) by writing m for = — ; C for %c s , and M G' for the " general " terms> so that, t = 2»o 1 (*oi-Rlog I ^| D ) +2» s (<#, s -Rlog I ^| u )+M G'. . (8) It will be convenient also to write G / for ^ — (M GT), the "general ' terms in s~^ ; similarly GJ for those in |i, etc. dn s Now, in the first place, we have to show that it is permis- sible to assume that (8) has already been so arranged that Gr', G s \ Goi', etc., all vanish in the limit when C becomes Molecular Thermodynamics. 235 very small. This is essential to the rigour of the demon- stration that the Raoult-van't Hoff laws are still the limiting laws o£ dilute solution when the solvent is complex. Now G / is, at constant temperature and pressure, a func- tion of c i, c 02 , . . . . , as well as of e l9 c 2 , Owing to the chemical equilibrium controlling coi, r 2, • • • • , liowever, these quantities have, in the pure solvent, values depending only on temperature and pressure, and the departures from these limiting values, caused by the presence of solutes, will clearly decrease with the concentrations of those solutes. Thus as C diminishes, the ranges of variation to be considered of the variables cbi, C02, «... are progressively limited, the same being obvious in the case of c u c 2 , But clearly any finite, continuous, ditf'erentiable function of several variables must behave as a linear function if the range of variation considered of every variable is sufficiently limited *. Thus LtG' = 2c iZoi + 2cA 5 ( 9 ^ C-yO i. e. Lt(M G')=Woi+2W,, • • • • C 10 ^ where Z 01 , , ? 1? , depend only on temperature and pressure, being, in fact, limiting values of Gr 01 ', , G/ , respectively. But clearly 7 i» > ^> , can be transferred to, and included in the " linear " terms, — in O1 , , fa, re- spectively — whereupon the residual " general " terms will satisfy the requirement that G', G s ', G 01 ', etc., should all vanish as C becomes very small. We shall assume that in (8) this adjustment has already been carried out. Returning now to (7) and comparing with (8), we see that C>OO iU o 00 L OU 0l J = It ["icoiC^oi-R^g"^!] I = 0M ( saY ); ' ( U ) since, in the pure solvent, c ol , and 777 assume values dependent only on temperature and pressure. * Merely the obvious property of tangency in re-dimensions. The theorem quoted by Planck in treating simple solvents ['Thermo- dynamics' (Trans. Ogg), 1917, p. 225] appears to be the particular case of this, when the ranges of variation of the variables are all located (as here in the case of c 1} c 2 , . . . .) close to zero. 236 Mr. Bernard Cavanaffh i=> on Then we have but again remembering Sj£dta=0, (13) O n oi and so from (8) '^ffr*jXpn(~Bdlogg$fe) + fw«V (15) c=o c=o = <£ M + R(idlog(l+mC-t- jScoirfGoi. ■ (16) (17) c=o c=o Now (6) with (8) gives \B^/Mo L \<Wn J, ft H-mO And if we write (16) in the form ^^M+nfirfi.og (i+mC)+a M , . (is) *c=o it is at once clear that the " general " terms thus adopted (and therefore, of course, the " linear " terms similarly) are connected by the Gibbs fundamental relation [see note at end of this paper], for (M G') being a function of n 0l ^i , homogeneous and of the first degree, Z>ioidG ol ' +2n s dG s ' = ; i-e. -M Zc 01 dG Q1 ' + Xn s dG s ' = 0, i-e. Mo^Gm 4-2n,dG,' = 0*, . . , (19) which means that [M G M + %nJ3t s '} or (say) M G is, as a * The Gibbs relation might indeed have been used to obtain (16) direct from (17), <j) M appearing as the integration constant. The above treat- ment [(12) to (16)] appeared, however, to be more interesting, and to introduce M in a more natural and illuminating manner, in its relation to the original " molecular " expression for ^r. Molecular Thermodynamic 237 function of M and n^n^ , homogeneous and of the first degree. And the same will hold for the " linear " terms, (if (M G) be accepted as the new " general terms"), a point of obvious importance since the "linear"" terms must alone remain when the solution is " perfect." ^ retains the on s simple form (17), which may now be written (20) since d(M G)\ BG (21) G being a function of c x c 2 only (besides T and p), Also, of course, 122) \ B» S /noi ' and M G=M,G M + S«,G S (23) We observe, however, that this convenient arrangement involves theoretically a certain definite (though probably always very small) change in the division into " linear " and " general " terms, since M G' = 2h 01 G i' + n sG s / ; G'=Zc 0l G 0l ' + Xc s G s = G+f 2G 01 '^ 01 (21) ~c=o Xow it is readily shown that Grni' = d^oi i [n< v„ ^G v ^G n + m i G — ZCoi^-, 2.<?* ^— (25) and since, 2m 01 c 01 being unity, 2??z 01 <ic 01 is always zero, we have G '— g= (v ?^*- J 3<?oi c=o (20) To say that any such modification of the division into " linear " and "general" terms is due to, and represents, a departure on the part of the solvent molecules from " perfect " behaviour would be to make a qualitative statement of no 238 Mr. Bernard Cavanagh on practical significance. (24) is a quantitative statement, and the form to which it is reduced in (26) has practical meaning and value, as will be shown by means of a simple illustration in a short appendix to this section. From (26) it is clear that, owing to the relative smallness of the changes in c 01 , c 02 , produced by the presence of solutes, the difference between the "general" terms in the original " molecular " expression for -v/r and those in the " experimental " expression we have obtained will generally be very small. But it may have to be taken into account in making use of dynamical theory (at high concentrations). Of course, until and except when the "general" terms can be given more definite form, we cannot say anything about the way in which they will depend on the constitution of the solvent and its variation . For the present we have to suppose that c i c 02 are eliminated in terms of T, p, c 1} c 2 , s from Gr, which takes some form G = UXcp x f f x (c lC2 ) (27) Theoretically, and in the general case, the application of dynamical theory will precede this elimination of c 0] c Q2 , will deal in fact with the original general terms M G-', so that, in greater or less degree, knowledge of the constitution of the solvent and its variation will be necessary before such theory can join issue with practice, but in some cases, as the appendix will illustrate, this may not be necessary, even though the constitution of the solvent does affect the " general " terms. It should be noticed that the value and convenience of our " experimental " expression for -\|r is by no means entirely dependent on a deficiency of knowledge of the constitution of the solvent, though the latter makes it practically indispensable. In the " linear " terms the effect of the constitution of the solvent is concentrated in the quantity m, the mean molecular weight of the solvent. In the pure solvent this will have a limiting value m dependent only on temperature and pressure, and we can therefore regard the quantity ( — R log m Q ) as included in the quantity <f> s , when our " experimental" expression for yjr will finally take the form ^ = M [<£ M + R f = dlog (1+SC)] *c=o 4-Sn s [^-R{logc 5 -log(| +m c|]+M G . . (28) Molecular Thermodynamics. 239 when m is constant (at constant temperature and pressure) and therefore equal to m we get f=M„|> M + Jlog (1 +-, C)1 + 2wJ^-RJ logc,-iog(l + m O)T]+M G, . . (20) which is equation (52) of the first of these papers, from which the " second approximation " equations were obtained. It is, of course, not possible to say how, in the most general case, m will depend on the concentrations of the various solutes, but an interesting case, of probably very wide application, may be treated and will at the same time serve as an illustration. This is the case where m can be written, with sufficient approximation, as a series of ascending integral powders of C, the total solute concentration. This can be shown to be the case, for instance, when the various solvent-molecular species behave as perfect solutes (in the true sense, — not in the sense of the Raoult-van't Hoff laws). Some simple cases have been investigated, but the detail need not be given here. It will suffice to say that in the simplest case, for example where the solvent consists of two molecular species, m and (2 m ), the one the doublet of the other, we find that ~m can be expressed as m = m [l + 6{m C) + V (m C) 2 ], where the values of 6 and y depend, of course, on the proportions in which the two species are present in the pure solvent, but in any case cannot exceed §• and -3-g respectively (these maxima not being simultaneous). We may carry this expansion of m, which is formally convenient for our purpose, to one further term of which only the order of magnitude will matter, in .=l + 0(- Q C) + V (m Cy + Z(m C^ . . (30) the last term being, as we shall find, altogether negligible if £ is no greater than about y 1 ^. Approximating on the assumption that 6 and r) are of the 240 Mr. Bernard Cavanagh on order of magnitude ^ we obtain : log^+woO^aiO-iajC + JaaC 8 , . . (31) | J^log(l+mC)=C(l-ia 1 C + ia 2 C 2 -ia 3 C 3 ), (32) c=o where a 1 = ?w (l — 0), a 2 = m 2 [l-2(<9-7 7 )], ... (33) a 3 = m o 3 [l-3(0-77 + f it being clear that, as stated above, £ can be neglected altogether if no greater than about fa We get then the equations ( W =^ M +RC[i-KC+^ 2 c 2 -ia 3 ^] + g m j IV - U*° (34) |i- =^-R[log<; s -a 1 C + ia 2 C 2 -ia i) C 3 ]+G s and thence the successive stages : — U=<£ M +RC[Wmo(l-0)C + ; WC 2 ] +G M ) U/° (35) ^=* M + RC[l-^ O] + G M and at ,3M, II. . .- .... (36) ||£ = *.-R[logc.-m C]+G, and (|±=^ + RC + G M j I. * M ° ....... (37) f|£ =^-Rlogc, + G, \ 0^9 J I. and II. being the first and second approximations obtained by the preliminary treatment in the previous paper. Taking I per cent, as the " probable experimental error," Molecular Thermodynamics. 241 we get roughly the following upper limits (of total concen- tration C) for the applicability of the four successive stages of approximation : Approximation I M. 10 II 2M. Ill 5M. IV 8M. (38) — that is, considering aqueous solution, and assuming m to be about 40. The limits would be considerably different, par- ticularly in the case of III. and IV., if m were given a very different value, as can readily be seen. The practical criterion of a perfect solute in a complex solvent is now that its behaviour should be expressed by that one of the above succesive approximations appropriate to the total-solute concentration of the solution, with the "general " terms omitted. If the assumption of perfect behaviour in the case of a particular solute be made, an experimental determination of the quantities ra , 0, rj, etc., can be made and concordance in several such determinations made upon different solutes would tend to justify the assumption that perfect behaviour persisted up to the concentration at which concordance was found. According to the concentration reached (with concordance) some of the quantities m , 0, 77, etc. would then be known with some approximation (closest in m , next in #, and so on). Then, on the assumption of "perfect" behaviour on the part of the solvent-molecular species — that is, a sufficient approximation thereto, — these quantities would suffice to discover something about the constitution of the solvent. Thus two solvent-molecular species would be completely determined by a knowledge of m alone (that is, the propor- tions of the two kinds present would be determined), — and in this case the question of "perfect" behaviour of the two species would not enter. m and 6 would, suffice to deter- mine completely three species, or would provide a check if only two species were present, and so on. The problem, however, as a practical problem is compli- cated by considerations which are the subject of a succeeding paper, viz. the question of solvation and partial solvation of solutes. Finally, it is proposed to consider one point with regard Phil. Mag, S. 6. Vol. 44. No. 259. July 1922. R 242 Mr. Bernard Cavanagh on to the expansions of U and Y, the total energy and volume o£ the solution. (Fuller consideration is postponed, as this paper is already rather long.) In " perfect " solution we have U = %n 0l u 01 -f %n s u and ■„ } • • • ■ (39) + 2<n s v s J V = %n 0l v 0l Q = %n 01 q 01 + %n s q s , (40) where, according to the usage of the previous paper, Q is (U+joV), q s is (iis+pvs), etc., and we can write this Q = M 2coi?oi+2w,ft (41) 2coi#oi> however, depends (through c 01 , c 02 ; . . . .) upon the concentrations of solutes present, but has, in the pure solvent, a limiting value <^m depending only on T and p, and tc 01 q 01 = q M +%qoi\ dcoi . • • • (42) Jc=o = q M + %q 01 Ac 01 (43) And so Q = M ? M +2w^ + M SgoiAc i. . • ■ ( u ) Dilute now such a solution "infinitely" by adding a large mass M(/ of pure solvent at the same T and jo, for which Q' = M '^M. The united heat will be (Q+Q') = (Mo + MoO^M + Sw^ + MoSgoiAcoi. But since the solution is now " infinitely " dilute, we shall have on bringing it to the original temperature and pressure Q"= (Mo + M ')?m+2m^= (Q 4 Q / )-M 2?oiAc i. In other words, MqX^oi^^oi was the heat developed (evolved) on diluting the %n 8 molecules of solute. That is, there is a heat of dilution of ytq 01 Ac 0l (45) per gram-molecule of solute, in " perfect " solution in a " complex " solvent. The explanation of this apparent Molecular Thermodynamics. 243 anomaly is that in a "complex'' solvent the process of dilution is not quite so simple as in a "simple" solvent, being accompanied by a change in the constitution of the solvent — a reversion, in fact, to the constitution of the pure solvent. It is, of course, plain that M 2<7 01 Ac 01 , or SgfoAwoij is the ("chemical") heat evolution accompanying this reversion (at constant temperature and pressure). With (44) analogously we have U = M m m -» Xn s u s + M ti( Ac 0U . . . (46) V = M vm + Ws + M XiWW, • • • (47) and it may easily be verified from (LI) that (48) Appendix to Section III. A simple example, which is essentially merely illustrative, but may possibly be something more, will serve to make clear the practical significance of equation (26) concerning the slight modification of the " general JJ terms. In section V. of the first of these papers was obtained for the general term in the case of a dilute solution of a binary strong electrolyte M G'=RM <£V /2 , (49) where (// depended, in some way, upon a certain " effective" dielectric constant (D), which, at sufficient dilution, would be that of the pure solvent. Let us suppose that D depends on c s in such a way that <f>' is a linear function of c s 4>'=4>o'(l + ac,) 9 (50) and also, in the first instance, that this effect of c s on D is entirely independent of its effect on the constitution of the solvent, that is, that the slight change in the latter produced by the addition of the electrolyte would alone [if produced, for instance, by some different solute, c„ the concentration R 2 241 Mr. Bernard Oavanagh on of the electrolyte being made quite small] have but negli- gible effect on D, then G' is independent of the constitution of the solvent (practically), i. e. so that, i (Xoi =0 = 0; . . G = G / = UCJ> 'C^\1 + UC S ) 1 * G s = G-J = lR<£o' c * 1/2(1 +.5 aCg ) >)■'■' Gj\f= Zc 01 G i '=— 2^0 'cv 3 / 2 (l-f3ac s (51) (52) If, on the other hand, we suppose that the effect of c s on D is entirely dependent on the change produced in the consti- tution of the solvent, and would be fully obtained if the latter could be brought about in some other wa}', while c s was made very small, then we should have an( 2 dGT ^ Bcoi _ Rc 3/2 3<£! = Rc s 3 / 2 <£'a J c=o so that G'-a = |R(/) w s 5/2 (53) (54) (55) i. e. in this second case, G = G' -J 2|£'^ 01 =E</, V /2 (1+I«,) g s = g; = i r^ v /2 (i +«<■») (56) Gm = 2<"oiG -1 BG' d^oi c=o In both cases, of course, -<Z< -iRtf> V /2 (l + ^) G s = G,/, but they are not the same in the two cases because in the second case <f>' is a constant when w i> n 02 . . . . are held con- stant, but not in the first case. (For the same reason ScoiGoi' is a l- different in the two cases.) In the second case we note that although the constitution Molecular Thermodynamics. 245 of the solvent enters into the "general" terms it does so only through the quantity D, and if this is the dielectric' constant of the solution in bulk it can be. measured and so determined as a function of c s without considering the corre- sponding variation in the constitution of the solvent, or the way in which the latter exerts its effect on D. If, as is probable at the less extreme dilutions, D is not the experimental or " bulk " dielectric constant, but a certain statistical-average quantity of a peculiar kind, then its varia- tion is, at least partially, not due to a variation in the solvent, but directly to variation of c s as in the first case above. Clearly from the preceding it would theoretically be pos- sible iu such a case to determine from comparison of theory with experiment whether the effect of c s on D was direct or indirect or in what proportion both, but it might not be practicable owing to the smallness of the effects to be measured. Note on the " Gibbs Fundamental Relation." Consider any property it of a homogeneous substance or phase, which is determined in magnitude by the composition of the phase and the quantity of substance considered. In view of the homogeneity it must then be proportional to the quantity of substance when the composition is fixed. Such properties are (at constant temperature and pressure) U and V, the total-energy and volume, Q or (U + pY), which might be called the "reversible heat content," and any thermodynamic potential such as entropy, free energy, Gibbs' " chemical potential," or Planck's yjr, which may all be expressed (at constant temperature and pressure) as func- tions of the quantities ]M\ M 2 .... of the constituents which suffice, under the conditions considered, to produce the phase. We can show, as Planck does in the case of i/r, that for any such property it, - = SM W • • • • • («) for if e be some infinitesimal fraction and we remove (eMj) of the first constituent, clearly it is diminished by eMx ^rr- . Removing simultaneously the same fraction of the total quantity of each constituent we diminish w, in all, by SeMx^r^-, or eSMj— ^. But, in so doing, we have 246 On Molecular Thermodynamics. simply removed the fraction e of the whole phase, without altering its composition, so that tt must have diminished by en, that is whence (57) follows. When tt is either yjr, or Gibbs' "chemical potential," a whole system of phases in equilibrium can be considered together, since then ^^ , etc., are the same in every phase. Without actually quoting Euler's theorem, Planck remarks, in regard to yjr, that this relation means that yfr, as a function of M : M 2 . . . ., is homogeneous and of the first degree, though not, of course, in general linear, and the same remark applies to our typical property tt. From (57) we can at once get a more practically useful relation by differentiating both sides fully:-— that is, MS)* (58) Now equation (97) of Gibbs' classical paper reduces at constant temperature and pressure to Xm^dfjbi = and is then simply (58) applied to Gibbs' " chemical potential." (58) is therefore referred to in these papers as the " Gibbs fundamental relation," but its general applicability to any property of the type of it (for a single homogeneous phase) is to be borne in mind. It is to be observed that while the constituents whose masses are M 1 M 2 .... must be sufficient to produce the phase under the conditions considered, they need not be all necessary — they need not be the " general-thermodynamic " components. And since also (obviously from the form of (58)) MiM 2 . . . . need not be expressed in the same units we see that equally valid is the form w (£) = 0, (59) the " molecular-thermodynamic " form, in which n x n 2 . are the numbers present of the various molecular species. The Calculation of Centroids. 247 It is an important point in the treatment of the two problems, "complex solvents " and "solvation," presented in this and a succeeding paper, that in the " practical " or "experimental" expression for yfr finally obtained the "linear terms" by themselves satisfy the Gibbs fundamental relation, for in perfect solution these terms alone remain. And this is preserved in the successive approximations. The relation also serves as a useful check upon the correct- ness of the detail. Balliol College, March 1922. XVII. The Calculation of Centroids. By J. G. Gray, D.Sc, Car gill Professor of Applied Physics in the University of Glasgow *. THE position of the centroid of a plane arc or area is usually determined by the application of one or other of the two theorems of Pappus. The methods described and illustrated below seem, however, to be novel ; they are useful in a great number of cases, including many to which the theorems of Pappus do not apply. Fi<r. 1. Fig. 2. Consider a system made up of two masses M and m (fig. 1) . Let the centroids of m and of the system M and m be at a and G respectively. Now suppose the mass m moved so that its centroid is brought to a'. G moves to G', where GG' is parallel to aa' ; and we have (M + m) GG' '= m aa' . As a first example, consider the case of a circular arc AB mass per unit of length m (say). Let be the centre of the circle of which the arc forms part. Now suppose the arc * Communicated by the Author. 248 Prof. J. G, Gray on the rotated about in its own plane through a small angle 0, so that A is brought to A' and B to B'. In effect the small portion AA' of the arc is transferred from one end of the arc to the other. The mass of this element is mrd, and it has been translated (virtually) through the distance 2r sin a/2, where a is the angle AOB. If G is the centroid of the arc, we have obviously mrO 2r sin a/2 = mra OG 0, or 0G = 2r sin u/2 As a second example we take the case of a sector of a circle OADC (fig. 3). Let the sector be turned in its own plane through a small angle 6 about an axis through 0, so that A comes to A', B to B\ The effect of the rotation has been (virtually) to transfer the triangle OAA' to OOC. Fig-. 3. The centroid G of the sector has moved parallel to gg' through a distance OG 6. The mass of the sector is JrW, and that of the triangle OAA' is ^r 2 6cr, where a is the mass of the sector per unit area. Since aa' = k t sin « , we have 99 -r 2 6<r - r sin - = - r 2 u<r OG 0, or OG = 4 r sin a/2 Again, let it be required to find the position of the centroid of a segment of a circle ABC (fig. 4). The segment is turned in its own plane, about 0, through a small angle 0. A is thus brought to A f and to 0\ If the mass per unit of area of the segment is cr, the mass of the triangle DAA' Calculation of Centroids. 249 is o-(9^?'sin|a rsinja, or ^o-tfr 2 sin 8 %ct. The area o£ the segment is |r 2 « — r 2 sin -|acos J«, and its mass is cr/' 2 (-|a — sin \ol cos Ja). Since <7</ is v|?'sin-^«, we have o-? ,2 (-|« — sin Ja cos \c*)x6 = §o-0?< 3 sin 3 £a, or n a? = ?' sin 3 i« 3 ^a — sin ^acos-^a* where a? is the distance of the centroid of the segment from 0. Consider next the solids obtained by dividing a right circular cylinder into two parts by means of the plane abed (fig. 5). Let it be required to find the position of the Fis:. 5. centroid of the lower solid. We suppose the solid rotated through a small angle 6 about the axis 00' (the axis of figure of the complete cylinder) ; a is thus brought to a', b to b', c to c', and d to d' . In effect the wedge ebb'e'cc' has been removed from the solid and replaced in the position eaa'edd'. If A A denotes an element of area in abed at a distance x from ee\ the volume swept out by this element due to the turning of the solid is AAxO. The mass of this element of volume is p/\Ax0, and since the element is moved (virtually) through a distance "2x, we have, if V is the volume of the solid, YpOUe = 2pdZ AAx\ where the summation is made over the complete area abed. Hence V x OG = AK 2 , where A is the area abed, and K is its radius of gyration 250 Prof. J. G. Gray on tin about ee'. If a is the angle bOa, and I the length of the solid, we have for the sectional area \t 2 {2it — ct^ + r 2 sin -^a cos \a. And since A = 2lr sin ^a, and K 2 = %r 2 sin 2 -Ja, we have OG r sin 3 -ia IT- ^a + sin ^acos \a Similarly, if G' is the centroid of the upper solid, we have ia — Sin fa COS fa The positions of the centroids of the solids obtained by dividing a right circular cylinder into two portions by means of planes Oa, Ob (fig. 6) are easily determined. If G is the centroid of the larger solid, we obtain at once, by supposing Fig. 6. the solid turned through a small angle 6 about the axis of figure of the complete cylinder, so that a arrives at a', and b at &', . VxOG = 2AK 2 sini<*, where A is the area represented by Oa, and K is its radius of gyration about the axis of figure. We have, if I is the length of the cylinder. OG 2ZrJr 2 sin£a P(2fl—a)7 4 r sin ia 3 2ir-cc ' If the cylinder is divided into two equal parts, we obtain Calculation of Centroids. 251 the position of the centroid of each by putting- a = 7r. Thus 0G=^. O 7T For the portion of a sphere shown by the firm lines of figure 7, we have, if OG is the distance of the centroid from the centre of the complete sphere, where A is the area abed, and K is its radius of gyration about its diameter. Thus OP — 71 " r2 S ^ llS 2 a 4 r2 Sni2 2 U 3 r For the portion of a sphere enclosed by the firm lines of figure 8 we have ,2 \r 2 sin | x = •nr 4 *?' sin ka. o 2lT — OL 7TT 6 3 2tt 3 irr sin \a. 3 27T — OC where x is the distance of the centroid from 00'. For a hemisphere a = 7r, and x = i r. Finally, for the portion of an anchor ring shown in figure 9 we have, if r is the radius of the ring and a the radius of the section, OP — 7ra,2 (i a 2 + 7 ' 2 i ^ sin £* 7ra J X 27rr X Z7T _ 2{r* + ja*) sin Ja y(27T~a) ' in which reduces to 2rjir when a = 7r and a is very small comparison with r. If a body is floating partly immersed in a liquid, the 252 Messrs. Trivelli and Righter on Silbersteins distance of the centre of buoyancy B (the centroid of the displaced liquid) from the metacentre M is given by BM = AK 2 /V, where V is the volume of the displaced liquid and AK 2 is the moment of inertia of the water-line area about the intersection of the wedges of emersion and im- mersion. The equilibrium of the floating body is stable if Fig. 8. Fig. 9. M is above G, the centroid of the floating body, that is if BM > BG ; it is unstable if BM < BG ; and it is neutral if BM = BG, that is when M coincides with G. In figure 5 we may suppose the complete cylinder floating in water so that abed is the water-line area. The cylinder is obviously in neutral equilibrium so far as turning about the axis 00' is concerned. Thus M lies in the line 00', and we have 0G=^ University of Glasgow, May 1,1922. XVIII. Preliminary Investigations on Silbersteins Quantum Theory of Photographic Exposure. By A. P. H. Teivelli and L. Righter*. Introductory. THIS paper is the first of a number of investigations now being conducted in this laboratory to test experi- mentally the light-quantum theory of photographic exposure recently proposed by Dr. Silberstein before the Toronto meeting of the American Physical Society. * Communication No. 141 from the Research Laboratory of the Eastman Kodak Company. Quantum Theory of Photographic Exposure. 253 Originally, these experiments were started independently of that theory, onr intention being primarily to study the effect of the clumping, or clustering together in groups, of silver halide grains in a photographic emulsion *. According to Slade and Higson f, " It seems reasonable to assume that each grain acts quite independently and that one grain which has become developable is unable to make a grain, situated in close proximity to it, developable unless the latter grain is developable in itself." From the table (p. 256) it is readily seen that this statement is not true, but the grains when clumped together act as one grain for development to the limit. Since each clump acts as one grain a very much broader range of grain sizes, or their equivalent, is obtained extending from the smallest single grains up to the largest clumps (containing 30 or more grains) in a given emulsion. It was thought that these results afforded a rigorous test for Silberstehrs theory, and it seemed therefore worth while to compare them with the implications of that theory. Silberstein's fundamental formula is essentially, i. e. apart from chromatic complications, and disregarding the lateral dimensions of the light quanta, & -j -na where N is the original number, per unit of the p'ate, of the class of grains of size (area) a, n the number of incident light-quanta, again per unit area of plate, and k the number of grains affected J. If the finite cross section, a of the light-quantum is taken into account, a is to be replaced by '-['Va*- At first, the rapid increase of k with the size a as required by that formula seemed (qualitatively speaking) not only attained but even exceeded by the experimental results. This seemed to us to indicate that the sizes (areas) of the clumps of grains were under-estimated by us. In fact, for * Extensive experiments are being conducted in this laboratory to study the clumping of grains in different concentrations of the same emulsion. These investigations will be published later. t Slade and Higson, "Photochemical Investigations of the Photo- graphic Plate." Proc. Roy. Soc. xcviii. p. 154 (1920). % Silberstein, L., "Quantum Theory of Photographic Exposure,'' infra, p. 257. 254 Messrs. Trivelli and Kighter on Silbersteins an estimate of the areas of all clumps one and the same average grain size (area) had been assumed throughout. Upon recalculating the results, however, and assigning the correct average grain size (area) to the single grains and to the different clumps, the very interesting fact was observed that the average grain size (area) increases from the single grains to the clumps of two, three, etc. The corrected results conform, even better than expected, to the above formula, with a finite a. Experimental. These experiments, although of an extremely tedious and trying nature, were performed with the utmost care and, to the best of our knowledge, all sources of error were either eliminated or reduced to a minimum. Only a brief descrip- tion of the experimental procedure will be given at this time, as a more extensive paper containing further experi- mental results is to be published in the near future with a detailed account of our methods of photomicrography, and in which all errors will be discussed fully. A simple silver bromide emulsion was used for these experiments having a speed of 112 and gamma 0*8 for six minutes development in an ordinary pyro-soda developer at 17° C. The average size of the grain is about 09yit diameter. The method of preparation of strips for sensitometric exposure is, briefly, as follows : — One 5 in. x 7 in. plate of the original emulsion is soaked in distilled water for one half-hour at 0° C. to 8° C. (All work for sensitometric exposure was done in a dark room by the aid of a dull red safelight, Wratten Series 2). The water is removed and a warm solution of gelatine, alcohol, and water is added and the whole solution heated in an oven for 20 minutes, while care is being taken not to heat over 10° C, because above this temperature much fogging takes place. With several such applications of the aforesaid solution the emulsion is entirely removed from the plate and the resulting solution made up to such a volume that it will give one layer of grains upon coating and drying. Some of the slides are used at once to get the clump frequency data. Those for exposure are backed with an opaque substance to prevent reflexions, then exposed in a sensitometer, developed to gamma infinity with a pyro-soda developer at 17° C, washed, and the developed silver removed with a dilute solution of chromic acid and Quantum Theory of Photographic Exposure. 255 sulphuric acid. The strips thus obtained contain the unde- veloped grains, and by taking the difference the number of: developed grains is calculated. The data given below cover only the first or highest density step of a Hurter and Driffield sensitometric strip *. 20 fields on each of 3 strips are employed to determine the developed grains. To determine the number of grains and clumps in the original one grain layer plate before sensi- tometric exposure and development, 10 fields on each of 4 strips were used. By taking this large number of fields on several strips we obtained a much better average. The results in both the above cases are reduced to a number of grains or clumps per square centimetre of one grain layer plate. Then as the dilution is known, one may, with certain restrictions, refer back to the original plate. All photomicrographs were made at a magnification of 2500 diameters and these negatives enlarged 4 times in printing. On the prints the grains and clumps are measured and counted, and then classified in class sizes (areas). The class sizes (areas) are to 0*2 /* 2 , 0'2 to 0*4 yu 2 , 0*4 to 0"6//, 2 , etc. The light source is a point source from a Pointolite lamp which is screened with a YVratten (H) blue filter to restrict the wave-length range and therefore increase the resolving power of the microscope. A cell containing copper sulphate solution absorbs heat rays and a cell con- taining a solution of quinine bisulphate excludes the ultra- violet light. The optical system is built up as follows : Cedar oil immersion condenser and objective, aplanatic condenser of numerical aperture 1*4, and Bausch and Lomb objective 1*9 mm. numerical aperture 1*3 in combination with a No. 6 compensating ocular. In the following is a table of our results. Column 1 contains the number of grains in each clump, column 2 the average area of grains in corresponding clumps, column 3 the number of grains times 10 -3 per square centimetre of original one-grain layer plate, column 4 the number of grains times 10 ~ 3 per square centimetre of developed one- grain layer plate, and column 5 gives the proportionate k number ^ of clumps affected. * The five remaining steps each corresponding to one-half the exposure of the preceding are now being counted and mapped out, and we hope to be able to publish the results obtained with them in the course of one or two months. 256 Quantum Theory of Photographic Exposure. Number of N.10" 3 per £.]0~ 3 per 2 sq. cm. 1 grain sq. cm. 1 grain k in Clump. P • layer plate layer plate W (original). (developed). 1 0-75 4 65770 1086-0 0*165 2 1-92 5 1322-0 594-0 0-449 3 3-03 664-0 508-6 0-766 4 4-88 1 328-0 2863 0871 5 -r.^ 9 i unreliable 249 9 157-8 240-0 155-0 0-960 0-982 6 7 8-6 1052 105-2 1-000 8 9-8 110 526 52-6 52-6 52-6 1-000 1-000 9 10 120 39-5 39-3 1000 11 26-3 39-5 26-3 39-5 1-000 1-000 '2 13 19-8 19-8 6-5 13-2 13-2 132 6-5 3-3 198 19-8 66 132 132 6-5 3-3 G-6 1-000 1-000 1000 1-000 1-000 1-000 1-000 1-000 14 15 16 17 18 19 20 21 6-6 13-2 13-2 33 1-000 1-030 22 23 3-3 1-3 1-000 24 1<3 6-6 1-000 25 66 1-3 1000 ■ 26 1-3 1-000 27 33 1-000 28 3-3 1-000 29 1-000 30 1-3 1-000 31 1-3 1-3 1-000 32 24-0 1-3 1-3 1-000 33 25-0 1-3 1-000 The agreement of the numbers of the last column with those calculated by Silberstein's formula (with N = 0*572 and <r = 0*0973), as given in his paper, is manifestly a very pronounced one. The differences between the observed and calculated values are even in the case of the 3 grain clumps entirely within the limits of experimental error, particularly concerning the area measurements. In continuance of this work the theoretical formula will be subject to further tests, the results of which will be published shortly. Rochester, N.Y. January 24, 1922. [ 257 ] XIX. Quantum Theory of Photographic Exposure. By L. SlLBERSTEIN, Ph.D.* 1. rriHE purpose of the present paper is to describe a first JL attempt at a light-quantum theory of photographic exposure, or of the production of the so-called latent image, with the immediate consequences of such a theory and some of its experimental tests. The silver-halide grains of an emulsion spread over a plate or a film base may be considered (apart from the smallest grains) as small flat plates, of comparatively small thickness, which in a dry emulsion lie almost parallel to the base. The sizes a (areas) of these plates range from submicroscopic ones up to 18 or 20 square microns. Even the most uniform emulsions obtainable in practice consist of grains of different sizes, the distribution of sizes among the grains being in each case characterized by what is technically called " the frequency curve " of an emulsion. In what follows the number, per unit area of the photographic plate, of grains whose areas range from a to a -{-da will be denoted hyf(a)da 9 and a photographic emulsion will be shortly referred to as being of the type /(a). For certain emulsions /(a) is, with good approximation, an exponential, for others a Gaussian error-function of the area a, and so on. 2. Without, for the present, dwelling any longer upon details of this kind, we may pass at once to our main subject. According to Einstein's well-known hypothesis of 1905 light does not consist in a continuous distribution of energy, as in the classical theory, but is entirety split up into light quanta or discrete parcels of very concentrated monochromatic light, each parcel containing a quantum of energy, hv = h\/c, in obvious symbols. Somewhat more generally we may assume that only a fraction olE of the total light energy E is thus split into concentrated parcels, the remainder E = (l — ct)E being distributed continuously t> without however prejudicing the possibility of E being zero. Then, if E be the energetic value of a monochromatic * Paper read December 28, 1921, at the Toronto meetiug of the American Physical Society in affiliation with Section B of the American Association for the Advancement of Science. Communication No. 139 from the Research Laboratory of the Eastman Kodak Company. t Somewhat as in E. Marx's theory of " concentration places " or of " light specks'' as suggested by Sir J. J. Thomson. Phil Mag. S. 6. Vol. 44. No. 259. July 1922. S 258 Dr. L. Silberstein on a Quantum " exposure " of wave-length X, the number of light-quanta contained in it will be n= Tc X (1) From a recent conversation with Einstein, there are weighty reasons for making a=l. But since we do not prejudice its value, there is no harm in retaining this coefficient in the formula. Now, let us assume that the necessary and sufficient con- dition for a silver-halide grain to be affected, i. e., to be made developable (entirely or in part) is that it should absorb one light-quantum. Moreover, let us assume that a grain * does absorb a light quantum whenever it is fully hit by one, of a sufficiently high frequency v c or of a wave-length not exceeding a certain value X c . There are perhaps some experimental hints or more or less good reasons for making these two assumptions, but we need not stop to consider them here. It will be time to reject or to modify them when they are contradicted by photographic experiments. Nor is it necessary to enter here upon the mechanism by means of which a silver-halide grain is affected by a light-quantum, whether it be the knocking out of an electron, as suggested by Joly, or something entirely different. For none of such details will influence our main argument, to be treated in the next section. Only when we come to consider the dependence of the photographic effect upon the wave-length will it be interesting to con- sider the photoelectric hypothesis and necessary to take account of the fact that a photo-electron is not liberated unless the frequency exceeds a certain, the so-called critical value. Under these circumstances A e appearing in our second assumption will stand for the critical wave-length as known from Photo-Electricity. Again, whether a grain being " affected " is made developable in part only or throughout its whole area (no matter how large) is, in view of the* kind of the contemplated experimental tests, of no great importance. As a matter of fact, however, there is good evidence that a grain is always made developable as a whole, no matter what its size, and this seems even to hold for " clumps " or aggregates of several smaller grains, as will be explained hereafter. If so, then our formulae, to be "developed presently, will give not merely the number of affected grains but also, by integration, the total " mass " made developable and hence also the photographic density. But we may as well remain content with the formulae for the number (~k) of grains affected, and count these in all experimental tests. This, far from being a disadvantage, will enable us to subject the proposed theory to more precise, though at the same time more severe tests. One more remark. It will be understood that when we come to adopt the photo-electric hypothesis, a grain " affected " will stand for a grain * Or perhaps, more generally, one of every p grains hit, where p is a number to be determined by experiment, but presumably equal to unity. Theory of Photographic h\vposure. 259 which as a whole has been deprived of even a single electron only. It need not lose more than one electron in order to be made entirely developable. According to Professor Joly's original hypothesis * the latent image " is built up of ionised atoms or molecules." In our con- nexion, this does not mean that for every pair of atoms, say Ag Br, there is one electron liberated. Since every grain of silver bromide (as well as of silver chloride) is a crystalline, to wit a simple cubic space-lattice arrangement of Ag and Br atoms t, we may as well con- sider the whole grain as a single molecule. Such a crystalline structure being hit by a light-quantum and deprived of but a single electron, may well become susceptible throughout to the subsequent action of a developer. 3. With the assumptions just made the question is reduced to a mere problem in probabilities. Consider first the ideal case of equal grains. Let there be upon an area S of the photographic plate (feay unit area) N grains. Let a be the size (area) of each of them divided by S, and let n light-quanta impinge upon S, due allowance having been made for those which may be reflected or absorbed by the gelatine. The problem consists in finding the number k of grains hit and (if p = l) affected by this light exposure. Roughly, this simple problem can be treated as if -A 7 , k were continuous quantities, in the following way, familiar from many other instances (" mass-law "). At an}^ stage the number of unaffected grains is jST—k, representing an available fraction a(N — k) of the total area S. Thus, if further dn light-quanta be thrown upon S, and if their trans- versal dimensions be negligible as compared with those of the grains %, the corresponding increment of lc will be dk = a(]S r — k) dn } and since k = for ?i = 0, this gives at once k=B(l-e- na ), (A) which in fact will presently appear to be correct enough except for the practically unimportant cases of small iV or small JS r —k. More rigorously, but provided always that JV at least is a large number §, the required formula can be obtained by the * Nature, 1905, p. 308. t R. B. Wilsey, Phil. Mag. xlii. p. 262 (1921). % An assumption which will be given up in the sequel. § 'Which will be the case if S is taken large enough. Since plates and films in actual use contain as many as ]0 r) grains per cm.' 2 , S can be made as small as one-hundredth mm. 2 , and even less. S2 260 Dr. L. Silberstein on a Quantum following reasoning. The total area of silver halide being a fraction Na of the area S of the plate, each of the nJSa light-quanta will fall upon some grain. Of these nNa quanta one, say the first, will hit one grain, the next ^ N . quanta will fall upon another grain, the next _ will hit yet another, and so on, up to ^ 7 _ , .. quanta for the Irth grain *, Thus the required relation between k and n will be _1 1 1 l_ an ~ JX + N-l "" + N-k + 2 + N-k + r Now by a well-known theorem of Analysis J + i + l+ •••• +-=0 + logm + e(m), . . (2) JL o nx where C is Euler's constant and 0<e(«i) < 1/m. Thus in our case lo gyZTk= an + £ where f lies between — 1/A 7 and l/(J¥—k) or practically, since iVis at any rate a large number, e<?= £ (ir-*)<^. Whence, fc=JV(l— *-**-£) (3) In practicable experimental tests (counts of affected grains of a given, narrow size-class) the role of the correction f whose value f=e(iV r — k) can at any time be found by (2), may become perceptible only when the contemplated grain class is near exhaustion. Thus, apart from such extreme cases, we have again, as in (A), the simple formula k = A\l-e- na ), (4) which, though approximate only, will turn out to be accurate enough even for moderate values of N. A thoroughly rigorous treatment of the probability problem valid for any numbers iV, ?i, seems to be the * It is scarcely necessary to say that statements such as U N/N— 1 quanta hit another grain " are to be taken statistically as relating to averages over a group of many trials. Theory of Photographic Exposure. 201 following one. Of the n quanta the total number falling each upon some of the N grains will be * m = Na n, JS r a being the total area of silver halide, with S as unit area. Thus the problem is reduced to finding the distribution of m quanta among JS 7 grains. Now, let p m (i) be the probability of affecting, in a single trial, a number i of the JV grains, by the m quanta. By a combinatorial discussion which may be omitted here, I find P»W = jy»(]y-i)V ( } where a mi may be most shortly described as identical with the number of ways in which a product of m different primes can be decomposed into i factors. These numbers, which will be known to many readers from combinatorial algebra, have the obvious properties a m i=a mm —l, for any m, a mi = for i > m, and satisfy the general recurrency formula which enables us to write down successively without trouble any number of them. Thus, up to wi = 10, we have the following- table which the reader may continue to extend at his leisure. Columns correspond to con- stant m, and rows to constant i. 1111111 1 1 1 1 3 7 15 31 63 127 255 5 LI 1 6 25 90 301 966 3025 9330 1 10 65 350 1701 7770 34105 1 15 140 2401 13706 76300 1 21 266 3997 37688 1 28 462 7231 1 36 750 1 45 1 We may mention in passing that any a m i can be represented by f «»••= n D'"'- (i) {i ~ 1)m+ (») ('- 2 >*"- ± (!)]■ But for any numerical applications the table will be found more convenient. * In a large number of trials of the same experiment. t Cf. e. g. Lehrbuch d. Kombinatorik by E. Netto, Leipsig, 1901, p. 170. 262 Dr. L. Silberstein on a Quantum By formula (5) we have for instance the probability of hitting but one grain Pm\ Ij = jy^-l -> which is an obvious result. The probability of hitting two grains will be i?m (2) = (2— i-l)(j\r-l) . jy*- 1 , which is (approximately) 2 m N times as large as the preceding probability, and so on. If m be kept constant, p m (i) will increase with growing i up to its largest value for i = m, if ra<iV, or for i — N, if m > N. But it would be futile to expect, on an average, that distribution to which corresponds the greatest probability *. For all other distributions have some generally non-negligible probabilities and these are by no means symmetrically spaced with respect to the largest one. The only reasonable way of determining the number k is to define it as the average of i taken over a large number M of trials. Out of these M trials a number M p m (i) t of trials will give each i grains affected, and the total number of grains affected in all M trials will be XM^p m (^), to be summed from i = l to i = m, if m-<N and to i — N\i m >iV r ; but since a m i = for i > m, we can as well extend the sum in each case from i=l to i = N. Dividing this sum by M we shall have the average number of grains hit in one trial, i. e., by (5) 7Vf N in - This, with m — Nan, is the required rigorous formula for the number of grains affected, i. e., hit once or more. In order to see how this complicated formula degenerates into (4), which, of course, will be our working formula, develop the sum in (6). Collecting the terms in i^ -1 , N~ 2 etc., and taking- account of the values of a m ,\ it will be found that l mij , 1 (m\ 1 fm\ ( 1 V" -1 * If, say, ?n<:N, the most probable distribution is the equipartition (a quantum per grain), corresponding- to p m (m)= T^ m /j^^_ \ t ■ This would give as the number of grains affected k = m = Nna, or just the first term in the series development of (4), which would be hopelessly wrong- unless mN were very small. t "With a deviation ruled by Bernoulli's law. Theory of Photographic Exposure. 263 or, dividing by N and subtracting from unity, '-i-»(t)(-»)*-*o(-ir- N i. e. ultimately -»-(»-ir m This is rigorously equivalent to or identical with (6) for any m and X Now, for large JN", and any m, equation (6') in which (1 — l/N)*=:l/e, asymptotically, gives at once k -™ -r T =l — e * = l — e^ va . This is the connexion between the rigorous arithmetical formula and the exponential one. It is needless to insist that under the conditions prevailing in all practicable experimental cases there is more than suffi- cient mathematical accuracy in formula (4) k = N{l~e- na ) J which, apart from minor modifications, will henceforth be used in what follows. This formula is of the familiar type proposed (1893) by Elder, with the notable difference, however, that while his exponent contained a free " parameter " or coefficient to be evaluated empirically and principally depending upon "grain- sensitivity " and wave-length, both coefficients in (4) are completely determined, and the exponent moreover shows an explicit and most essential dependence upon the size (a) of the grain, and in the right sense too, i. e., giving an increase of the " speed " with grain size. The comparison with experimental facts of Elder's and of a number of other formulae, constructed empirically, is too well-known to be discussed here * Suffice it to say that, although it repre- sents to a certain extent the photographic behaviour (the " characteristic " curve) of some emulsions, and particularly those with what is termed an " extended toe," it cer- tainly shows considerable deviations from the observed characteristic curves. Yet it will not be forgotten that all these comparisons bore upon the resultant total photographic densities, containing or integrating the effects of grains belonging to a broad range of sizes (a), instead of equal * Cf. for instance, a paper bv Dr. F. E. Ross, Jonrn. Opt. Soc. Amer. vol. iv. p. 255 (1920). 264 Dr. L. Silberstein on a Quantum grains, so that no better agreement could be expected. The refined experimental tests which are now in progress in this laboratory, and by means of which it is hoped to corroborate the proposed theory, deal, as they should, with separate size-classes of grains. But questions concerning the comparison of the theory with experiment will be treated in a later part of the present and in subsequent papers. 4. Passing next to the case of an emulsion of any type /(a), it can be easily proved that the approximate formula (4) will hold for each class of grains separately. In order to see this it is enough to consider the case of two distinct classes of grains. Thus, let there be N± grains of size a l and N 2 grains of size a 2 spread over the (unit) area S of the plate, and let n light-quanta be thrown upon S. Of these a number m 1 = niV r 1 a 1 will fall on the a r grains and a number m 2 = nN 2 a 2 upon the a 2 -grains. It remains only to be found how many ax-grains will be hit by the m 1 quanta, and how many of the a 2 -grains will be hit by the m 2 quanta. Now each of these is a problem of the kind we have already treated. The number k x of ax-grains hit will be given by 1 J_ 1 i^x + iVi-l^ + iV 1 -/r 1 + l- naiJ and similarly for the a 2 -grains, so that k x and k 2 will each be determined by the previous formula for k with J\ 7 , a replaced by JSr iy ai, and N 2% a 2 respectively. Similarly for an emulsion consisting of three or more classes of grains. Thus, also, for an emulsion of any type /(a), the number of grains ranging from a to a-\-da hit and (if £>=1) affected by n impinging light-quanta will be k a da, where, apart from the correction f The total area * of silver halide affected or made developable will be found by extending the integral K- \akada (8) over the whole range of sizes, say from a 1 to a 2 . If, for instance, f(a) = Ce~ fl - a , say from a = a 1 to a 2 , where C, ix are constants, as in the case of some films and plates * It will he kept in mind that a stands for the " efficient " area of a grain (plate), i.e. for the orthogonal projection of the grain upon the film base. Theory of Photographic Exposure. 2G5 investigated in this laboratory for their frequency curves, then, with A written for the total area of silver halide, But this only by way of illustration. The fundamentally important thing will be formula (7), applicable to each size-class of grains separately. .In fact the experimental verification of the theory now in progress in this laboratory deals, not so much with K but, as it should, microscopically, with k = k a for each class of grains separately, including clumps of grains. Before passing to a further discussion and development of the elementary formula (7), but one more remark concerning the presence of more than one layer of grains. The case of two or more layers will at once be seen to be reducible to that of a single layer. In fact, either a grain of, say, the second layer and of size a is not shielded by any of the first layer or else it is thus screened off and only a part b of it remains uncovered. In the former case the grain in question will simply be classified among those of size a of the first layer, and in the latter case among those of size b. This will hold with respect to the exposure to the impinging light- quanta, and b will also be the contribution of the grain in question to the photographic density ; for its covered part will remain inoperative. Similarly for three or more layers. In fine, the presence of a plurality of layers of grains will modify only the frequency curve N a ~f{a) which would otherwise belong to a single layer. We shall henceforth assume that this factor has already been taken into account in constructing the function /(a) or in microscopic counts of the grains within every particular size-class. We disregard here, of course, such factors as a possible absorption of light in additional strata of gelatine. 5. Dependence upon wave-length. — Once more return to the elementary formula (7) or (4). Denote by s the ex- ponent so that N L e ' Under the more or less implicit assumption that the trans- versal dimensions of a light-quantum are negligible in comparison to those of a grain, we had s = na. But it will 266 Dr. L. Silberstein on a Quantum be seen presently that such an assumption is too narrow and unnecessarily so. In fact, substituting the number n of light-quanta from equation (1), the exponent s* will become s = y-i?X, or, if we put for brevity f3 = ot/hc, . (9) which may be considered as a constant, s=/3E.a\ (10) Here E is the incident light energy (exposure) and X the wave-length of the light assumed to be monochromatic. Thus the sensitivity exponent would be directly proportional to the wave-length, and the number k would, for constant E and a, increase steadily with the wave-length of the incident light up to the photoelectric critical value X c and then drop suddenly to zero, k j r = l-e- const -\\<\ n k =0, X > X c . Now, such a sensitivity curve does not seem to resemble the familiar experimental sensitivity curves which show a more or less gentle maximum followed by a gradual decrease down to zero. It is true that such experimental curves | represent the resultant effect due to grains of a whole range of sizes, so that the k — X curve belonging to a single class of (equal) grains may well be of the said abrupt type, — a question to be decided only by micro-spectrographic experi- ments and counts now in progress. Yet it seems advisable even at this stage to provide for the possibility of smooth maxima preceding the critical wave-length X c . This might be obtained by attributing to grains of different sizes different values of X c . For then, although the curve of each grain class would end abruptly , the superposition of such curves ending over a range of different abscissae, might properly displace and smooth out the resultant maximum. The correctness of such an assumption (X c a function of a) can at any rate be tested by direct experiments %. * s/JE can be referred to as the u sensitivity exponent." t Apart from the fact that they are not taken for E= const. \ Preparations for such experiments are now being made in this laboratory. Theory of Photographic Exposure. 2G7 Another way is to take account of the possibly /im'^ trans- versal dimensions of: the light parcels, which may perhaps be comparable with those of the lesser silver halide grains. Let us assume, therefore, that a light-quantum, of suffi- ciently high frequency, becomes efficient in affecting a grain only if it strikes it fully, or almost so. To fix the ideas, let r be the equivalent radius of a grain, i. e., such that and similarly let p be the average radius of the transversal section of a light parcel (so to call the space occupied by a quantum of energy). Then the efficient area of a grain to be substituted instead of a, will be a' = 7r(r — p, 2 ~a 1— ^ , and we shall have for the exponent s, instead of (10), s = na'=/3Fa [l-^Tx, for p<r< \< : \„. (10a) and 5 = for p > r or X > X c . It remains to assume in a general way that p, which may be the average of section-radii different even for parcels of the same wave-length, is itself a function of the wave-length increasing with X, without prejudicing, however, the parti- cular form of this function. Certain easily ascertainable broad features of such a function and thence also of the resulting factor in s, £(\)=\[ (ii) will suffice to ensure a maximum of the sensitivity exponent between X = and X = X e . The value of X c itself may still turn out to depend on the size a of the grain and on its physical conditions as well. Every process which will make the liberation of a photoelectron from the grain (crystal) easier will lengthen X c . Part of the effect of sensitizing may arise in this manner. But questions of this kind must necessarily be postponed until some further experimental data are gathered. Of such a kind is also the question whether p (which, for a given X, may also extend over a whole range of values) attains at all the semi-diameter r of even the smallest of the actual grains, and whether the corresponding wave-length A entailing the vanishing of s, exceeds or is smaller than X c as derived from direct photo- electric experiments. In absence of all knowledge concerning 268 Dr. L. Silberstein on a Quantum the spatial properties of light-quanta it would be utterly unjustified either to deny or to assert that their lateral dimensions are at all comparable with those of a silver halide grain (of the order of one-tenth up to several microns) *. If, by way of illustration only, p is propor- tional to a power of \, say p = b\ K , the only condition for the existence of a maximum of $(X), and therefore of sensitivity, will easily be found to be k>0. If this be satisfied, the maximum will occur at a wave-length \ m given by increasing with the diameter of the grain and bearing to X the fixed ratio ^=(2*;-r-iy/\ As a matter of fact, the maximum sensitivity is known to shift (by two or three hundred A.U.) towards the red by making the grain coarser. But thus far too little is known of the quantitative aspect of such an effect to entitle one to con- sider the above equation as anything more than an illustrative example. The precise form of the function p = p(\) can only be derived from spectrographs experiments followed by microscopic grain counts, or if arrived at by a guess, has to be verified by them. Such experiments are now in preparation in this laboratory, and their results will be reported in due time. A shift of the maximum sensitivity towards the red or the infra-red can, of course, be brought about by a function form more general than a mere positive power of the wave-length. 6. Generalities, and 'preliminary account of experimental tests. — The chief and most immediate consequence of the proposed theory is the essential dependence of the propor- tionate number of grains affected, k/J¥, on the size a of the grain, viz., the rapid increase of the former and, therefore, of " the speed " of an emulsion with the latter. Now, it has been known for a long time that (cceteris paribus) the speed increases notably with the size of the grain, and we shall see from the experiments to be described presently to how * According to E. Marx, Annalen der Physih, xli. pp. 161-190 (1913), the volume of a light-parcel, which according to him is only a u concen- tration place " within a continuous distribution of energy, is proportional to X 4 and amounts for D-light to almost 8.10 -7 cm. 3 , which even with a length of 10 cm. (200,000 D- waves) would still give a section area 8.10" s cm. 2 , just of the order of about the average grain area. There is, of course, nothing cogent about Marx's estimate, yet the matter is not without interest. Theory of Photographic Exposure. 269 large an extent this is actually the case. But perhaps the most tangible proof of the essential correctness of the assumption of spatially discrete as against continuous action*, seems to be the mere fact, disclosed by microscopic counts, that out of a number of apparently equal grains subjected to a sufficiently weak exposure one or two are affected while the others, nay their next neighbours, remain perfectly intact. It would be in vain to ascribe to these survivors a greater immunity or indifference to light. For it is enough to protract the exposure a little to make them succumb in their turn. Now such a behaviour is most typical of rain as contrasted with flood action, and the discrete light-quanta, hitting now this and now that grain, appear to be a most natural inference, while all attempts to bring into play the individual ''sensitiveness" of the units seem to involve considerable difficulties. As to the dependence of: the number of grains affected upon the wave-length, little more of interest in the present connexion is known than the qualitative fact of a shift towards the red of the maximum sensitivity with increasing size of the grain. Moreover, the available curves repre- senting the sensitivity across the spectrum concern the emulsion as a whole and not the separate a- classes of grain with which we are primarily concerned. Spectrographs and spectrophotometry experiments of such a kind, to be aided by direct photo-electric measurements, are now in progress in this laboratory, and all discussions involving wave-length will best be postponed until the results of these experiments and of laborious microscopic counts are forthcoming. Before passing to the mentioned quantitative test of the dependence on size, but one more general remark. The reader will have noticed the complete absence of the time- variable in all our formulae, the exposure entering only through the total number n of light-quanta or through the energy E which, in obvious symbols, is ^Idt. The pro- posed theory, therefore, as thus far developed, does not take any account of the little infringements against the reciprocity law t, iu short, of the so-called " failure of the reciprocity law." Now, it is by no means my intention to deny the * A rain as against a flood, of light, that is. t This early law asserted the dependence of the photographic effect (density) upon I and t only through the total incident energy or ex- posure \ Idt. For constant intensity this is It, whence the name of the law, relating to intensity and exposure-time as factors of a constant product. 270 Dr. L. Silberstein on a Quantum reality of these infringements which have been extensively studied and condensed into empirical formulae by Abney, SchwarzschiJd, Kron and others. But it has seemed inad- visable to encumber the very beginnings of the proposed theory by complicated details of such a kind*. The failure of the reciprocity law can more profitably be taken up later on, after the fundamentals of the theory have been somewhat solidified and extensively tested, and the prospects of mastering the "failure" theoretically are by no means averse, a very promising scheme seeming to lie in the possibility (suggested by Joly and taken up by H. S. Allen) of the liberated photo-electrons being regained by some of the grains which were deprived of them by previous impacts of light-quanta. In fine, the failure of the reciprocity law as well as the facts known under the head of " reversal " have at first to be neglected and considered as future problems for the light- quantum theory united with Joly's photo-electric theory, problems to which these com- bined theories seem well equal. To pass to numerical facts, a short description will now be given of the results of certain experiments undertaken in this laboratory by A. P. H. Trivelli and Lester Righter f which seem to corroborate the proposed theory most emphatically. In order to have a much wider range of sizes a than is usually afforded by the single grains, Trivelli and Righter applied their counts and area measure- ments to clumps of from one to as many as 33 grains, basing themselves upon the well- supported assumption that if one of the component grains be affected, the whole clump is made developable. (This, at anj r rate, is the behaviour * R. E. Slade and G. I. Higson, " Photochemical Investigations of the Photographic Plate," Proc. Roy. Soc. xcviii. pp. 154-170 (1920), on the contrary, make the failure of the reciprocity law their point of departure. They mention at the very beginning (p. 156) the possibility of a light- quantum theory and write the Elder-type of formula in I, t, remarking even that its coefficient would have a different value for each size of grain, but being discouraged by or rather preoccupied with the failure of the reciprocity law do not enter into the details of the probability problem, which would have disclosed them the structure of that co- efficient, and without much ado dismiss the quantum theory as '* impossible." Independently of Slade and Higson the possibility of a discrete theory (radiation in "filaments ") is mentioned by F. E. Ross, Astrophys. Journ. vol. Hi. p. 95 (1920). Dr. Ross, without being prejudiced against such a theory, notes even that it would lead ration- ally to a mass-law, but does not enter into the details of the probability problem and does not develop the theory. t For technical derails of these laborious experiments, see Trivelli and Righter's own note in this issue of the Phil. Mag. p. 252. Theory of Photographic Exposure. 271 of the larger, flat grains piled upon each other in part, although the smaller, spherical grains, in less intimate contact, may perhaps behave differently.) Such being the case, their experimental results should be covered by our formula with a written for the area of the whole clump, no matter how large and how numerous its components. This has seemed a rather severe test but the more so tempting and instructive. Since all the classes of clumps were given, in each trial, a unique exposure (through a blue filter specified toe. cit.) and there was no question of varying X, it will be most convenient to retain in the corresponding formula the original light- quantum number n as the parameter common to all clumps. Thus the formula to be tested becomes — =1 — e , s=na =na (»-»)'■ or somewhat more conveniently for computations, if o~ = 7rp 2 be the (average) area of the transversal section of a light parcel, lo g (l-4) = -™[l-^]\ . . (12) In the following table the first column gives the number of grains in a clump, the second the average area a of a clump in square microns, and the third column the per- centage of clumps affected out of all (i\ 7 ) clumps of each kind originally present, i.e. 100 k as deduced by Trivelli and Righter from their observations. Clumps of ain^. y oba> y c ^ Ay. 1 grain O-75-i 16-5 162 +0-3 2 grains 1-925 44-9 48'4 -3"5 3 , 3-03 76-6 68-9 +8-3 4 4-88 87-i 87-3 -0-2 5 „ Crls 96-" 933 2-7 6 ., 7-4-2 98-2 964 1-s 7 „ (8-6) 100 98-0 2-o 8 , (9-8) 100 99-6 0-4 9 , (11- ) 100 99-8 0-2 10 „ (12- ) 100 100 0-0 etc., etc. etc., etc. idem. idem. idem. 32 grains >24 100 100 0-0 33 „ >25 100 100 00 The most reliable a-values are those for the clumps of one and of two grains, being averages of the largest numbers 272 Dr. L. Silberstein on a Quantum of individual clumps ; the following a's are gradually less reliable ; from the 7-grain clumps onward the areas (bracketed) are only extrapolated, but since here y has practically reached 100, no greater accuracy is needed. The fourth column contains the theoretical values of y following from formula (12) with the constants, determined from observations 1, 2, 4, n = 0*572* per a 2 , l „=O097^, I ■'•■<«•> or p — 0*176//-. The fifth column gives A?/=y ob8 . — z/ ca i c> . The agreement is certainly very pronounced, the differences being, perhaps with the exception of the third, well within the limits of experimental error chiefly in the a-estimates. The fitting could be made even closer by retouching the constants n, a, but this is scarcely worth while, the formula itself being of a statistical nature, and the agreement being good enough as it stands. The same is manifest also from the figure giving a graphical representation of the last columns of the table. _fl - a oaoo — e — o — o o - o — o — ooo g ooooo c — e- — CALCULATED • OBSERVED 10 15 20 £5 The reader might think that the finite section cr, or whatever this parameter may stand for, has been forced upon the light-quantum and that tho observations might perhaps be as well represented with cr = 0, and another value of n. But actually, just the contrary has been the case, inasmuch as the author first tried the simpler formula log (l — k/fil)=—na, and then only found himself compelled to take in the correction factor as given in (12). In fact, Theory of Photographic Exposure. 273 dividing the observed values of log (1 — kjN) by the areas the reader will find that the quotient increases considerably and systematically, apart from a casual drop at the fourth clump, throughout the whole series of the clumps. Thus, the correction factor seems to come in quite spontaneously. On the other hand, there is nothing unlikely about the values of either of the constants (12 a). Our units of area being here square microns, we should have 57 millions light- quanta per cm. 2 (about which judgment has to be suspended until absolute energy measurements are available), and as the cross section of the space occupied by each of them (on an average) a little less than one-tenth of a square micron or a diameter of about 0*35 micron. Since each is presumably of about the order of a million wave-length long, they are rather slender at that cross section, and, instead of light parcels, as they were called above, would perhaps deserve rather the name of light darts. In Einstein's own theory there is nothing on which to base an estimate of the volume occupied by a light-quantum, but on Marx's less radical views this is about 8.10" 7 cm. 3 for D-light and proportional to \ 4 , and therefore in our case (narrow blue spectrum region with maximum at A=0'470yLt) about 3.10 -7 cm. 3 , which with the said cross section would mean a length of 3.10 6 //. or over six million wave-lengths. But this by the way only. The important thing is to see whether the above numerical value of the average cross section of blue light darts will continue to fulfil its function with regard to the remaining "steps" (weaker and weaker exposures) of plates coated with the same emulsion, the above being the highest " step." These have just been completed in this laboratory, ceteris paribus with the above one, and are now being subjected to counts and area measurements. This material will also serve to test the constancy of k/ N if, varying n and a', their product is kept constant. An account of the results of these and of several other experiments now in progress will be given in future papers. I gladly take this opportunity to express my best thanks to Dr. C. E. K. Mees for having proposed to me the problem of " discriminating, if possible, between the consequences of a discrete and a continuous exposure theory/'' and to my col- leagues Trivelli and Righter for furnishing the results of their experiments. Rochester, N.Y.. January 19, 1922. Phil. Mag. S. 6. Vol. 44. No. 259. July 1922. T L 274 ] XX. An Analytical Discrimination of Elastic Stresses in an Isotropic Body. By R. F. Gwyther, M.A* Sir G. B. Airy obtained from mechanical considerations (British Association Reports, Cambridge, 1862) a solu- tion in Cartesian coordinates of the mechanical stress- equations, but he ignored all elastic requirements. In one sense this paper may be regarded as an extension of Airy's scheme, though it has nothing in common with that scheme either in general plan or detail. The method is purely analytical, depending upon general solutions of the mechanical stress-equations and upon the development of a scheme for the selection of those stress- systems which satisfy the stress-strain relations, briefly called Hooke's law, from the general mechanical stress- systems. It is shown that the elements of a mechanical stress depend upon an arbitrary primary stress-system, and, to form a connexion with the stress-strain relations, I introduce a subsidiary, but allied, stress-system which is such that the vector system naturally deduced from it possesses inherent qualities distinctive of the displacement corresponding to an elastic stress-system. The main body of the paper consists in discussing the requirements necessary to ensure that the stress-system should be an elastic stress-system. The displacement, according to this method, becomes somewhat incidental, however necessary, and the elements of stress are given prominence. There are no displacement equations. In the first instance I deal with a body under tractions only, and extend the scope of the results later. I. Introduction. 1. By first treating certain ancillary matters as lemmas, the steps in the final stages can progress more steadily. The mechanical equations of stress in a body under * Communicated by the Author. Analytical Discrimination of Elastic Stresses. 275 tractions only are, in Cartesian coordinates, "d.v B*/ "dz 3.1* By B~ B* By + d~^ "' W and they are identically satisfied by the values B- 2 By^ByB*' B^l 3. Q = R_ ^.cW, *"" V 3*" T = U = B^'B- b 2 3 , 2 avs B* 2 B#B* 3#By' B^h _ j^h d« a B«By B 2 ^ 3 b 2 ^ av, B^B* B3/ 2 B 2 f 3 3y3-~' B 2 # 3 BVi B 2 ^ 2 B#B# B#B# ByB^ • • • (2) These contain six arbitrary and general functions, and form the general solution of (1). Lemma A. These six arbitrary functions have the same mode of resolution on transformation of axes as the elements of stress. For proof I use the method employed in the subject of differential-invariants. Imagine the axes of coordinates to be rotated about their own positions by the infinitesimal amounts co z , e» y , o) z , and consider the consequent changes in whatever quantities we may be considering — components of a vector, elements of a stress, etc. These changes will be linear functions of 60 x , ftty, (O z . T2 276 Mr. R. F. Grwyther on an Analytical Discrimination For example, in the case of the components of a vector u, v; iv, the changes will be : in u, —(D y w-\-co z v, in v, —cd z u + w x w, in w, —co x v-\-oc>yU. These changes will now be represented by partial dif- ferential-operators L\, 2 , Xl 3 arranged to produce the coefficients of co x , w v , co z in the quantities considered respectively. . Thus, for vectors generally, o 3 a This covers direction-cosines (Z, m, n), and may be made the basis of most operators. For example, we may deduce for stresses generally : ni = 2S (^-li) +(R - Q) l- u 9 V T ^ The differential coefficients A A A resolve as corn- er o t V o^ ponents of a vector, but, for simplicity, we must introduce a different notation. Write d x for B/B«, c? 2 for 3%, tf 3 for d/3*, and df„ for d 2 fdx 2 , d 12 for B 2 /d<^, d 22 for B 2 /3/ 7 and so on ; then for first differential coefficients of Elastic Stresses in an Isotropic Body. 211 as for vectors, and we deduce for second differential co- efficients Ql = , ' 1, M ! "'' l 'Bl', +2 '' !i trs)" (4 " 4, S' n$ = ^mt^ 3 ^hz + u ™ farr w~ (**-*«) §^ 2 - It is required to establish that in (2) 1? 6 2 , 3 , yfr lt i/r 2 , yjr 3 act as stresses. Actually I shall assume that the operators n i? Xl 2 > X2 3 act upon them as upon the elements of stress, and first examine this hypothesis when the three operators separately act on the six equalities in (2). Selecting the first equality in (2) and applying the operator Xlj on each of the two sides of the equality, then fljP = and ^6 2 ~d 2 0, 9o V*i d: 2 d/y 2 dyd- becomes 2 y*i , o3 8 f. , «,y «?.-<?,) 2 B 2 (^-^s) 9« 2 By 2 " ctyd^ d.yd^ B 2 -.&-£)* which is null, and the hypothesis is not negated. In fact, the hypothesis is not negated in any case. The argument is then as follows : — If we had written did, n 2 #> etc. in the equations, and had proceeded to find these 18 quantities, we should have 18 linear equations from which to find them. The solution is therefore unique, and cannot differ from that employed by hypothesis and found to satisfy the 18 tests. We shall therefore regard 1? 6 2 i @z-> ^1? ^2? ^3 as acting as elements of a stress, though the} T have not the proper dimensions. It is proposed to find the conditions which must exist between these primary arbitrary stresses in order that the elements P, Q, R, S, T, U may be elements of an elastic stress. 278 Mr. R. F. Gwyther on an Analytical Discrimination Note. — If, in the ordinary notation for strains, we give a, 6, c each one-half of" the usual value given to it, strains would follow the same laws of composition and resolution as stresses, and would therefore have the same differential operators. In this paper, I shall use a, b, c in this sense- that is, one-halt' of their usual. value. Lemma B. Except when we have reasons for keeping the expressions quite genera], it will suffice to limit the arbitrary stress- system to such stresses as have the co-ordinate axes as their principal axes. . " If in equation (2) the elements of stress, P, Q, etc., are made zero, the set of equations will then be recognized as indicating that 1 = e, 2 = f, 3 = g, ty 1 — a, ^—b, ^ 3 = c, where e, f, g, etc. are elements of strain arising from some arbitrary displacement. Hence on the right-hand side of equations (2) we may always replace 0\ by 0\ — e, 6 2 by 02— f, 0s hy B — g 9 "^i by "ty\ — a, yfr 2 by ty 2 --b, yfr s by t|r 3 — c. Consequently we may eliminate several sets of three functions, such as a/^, ty 2 , and i^ 3 , when some displacement is possible which makes, say, ^*i = a, ^ 2 = b, ^ 3 = c. Hence p-_^_^3 etc 3y3* , etc., which is the form given by Airy, is a quite general form of solution, although for the purpose of this paper the full form given in (2) is requisite until we have decided upon some particular set of axes. 2. The choice of a vector to represent the displacement, and the descriptive criterion of elastic stress. The mechanical stress has been represented in terms of an arbitrary stress-system, and it is possible and desirable to represent the displacement in terms of a similar stress- system. For this purpose I form a subsidiary stress-system, of Elastic Stresses in an Isotropic Body. 279 indicated by fl.+fl 2 + fl 3 a Ol +Oi + O* n 0l + 0* + Oz n 2 u 2 ' 2 3 ' This subsidiary system may be described as comple- mentary to the primary stress-system, in the sense that the two together form a hydrostatic pressure whose intensity is one-half of the sum of the principal stresses or one-half of the First Invariant of the primary stress-system. I shall form the assumed components of the displacement from the elements of this subsidiary stress in the manner of forming a force-system from a stress-system. Thus, I shall write 2nw^-2^-2^ + ^ { 1 + 2 -0 s ). . . (3) Forming the values of S, T, U from these on the elastic stress-strain hypothesis we have s = a^i ay* aVs BVi_BVi 'dy'dz ~dx~dy ~dx~dz ~dy 2 "dz 2 ' etc. On equating these to the values for the same elements given in (2), we find they require V s f = 0, V 2 t 2 =0, V s f 3 = 0. . . (4) Since {0 h 2 > $3» ^i> ^2? ^3} ac ^ on transformation of coordinates as elements of stress, it follows that the system must consist of a hydrostatic pressure and a general stress-system, each of the elements of which is a Spherical Harmonic. That is, WWe V 2 *i = 0, V 2 % 2 =0, V 2 %3 = 0. . . (5) This is the descriptive criterion of an elastic stress- system. 280 Mr. R. F. Gwyther on an Analytical Discrimination 3. Completion of the discrimination. The metric criterion. The remaining requirements of the stress-strain relations may be written \o% oyJ p +Q+B =(3»-„)g4; + ^). . . (6 ) which are to be completed from the values in (2) and (3). The first two only confirm the descriptive criteria of (4) and (5). The last leads to dj/0^ a^os d#dy J •im - n r 2 , „ » . f a^, d 2 2 ?i 2 dyo^ d^o^ oocdy J J and therefore to (3m + 7i)V 2 (0i + 2 + 3 ) lor a«/ 2 oz 2 oylbz d#d- d^'dyJ or (m+n)S7 2 <j) ( 0« 2 0«/ 2 O^ di/Oz 0*'02 0*02/ J • • • (7) which is the metric criterion and completes the dis- crimination sought for. This can be integrated, and gives <p in terms of the %'s and ^s_, of which we can always take the ty's to be null, when desirable. This completes the investigation for Cartesian coordinates under normal tractions only and with no inertia terms. With the conditions in (4), (5), and (6), the equations (2) give the general elastic stress- system, and these con- ditions discriminate an elastic stress-system from any other mechanical stress-system. of Elastic Stresses in an Isotropic Body. 281 4. The inclusion of inertia terms. We must now modify equations (1) by writing pii, pv, pib on the right-hand side, where u, v, to are to have the values given in (3). We consequently replace P by P — p{9< 2 + 6 z — 6 x )\2n, Q by Q-K& + 01-00 /2», R by R- /0 (g 1 + g a _g 3 )/2n, S by S + /^h/», T by T+/njr 2 /w, U by U + p^/n. With these alterations the equations (2) still hold good. In forming our criteria, we equate values found from (2) to values given by the stress-strain relations deduced from (3). In these latter P, Q, R, S, T, U are to have the original values of these quantities and not those which replace them as above. Consequently, as our first step in the criteria, in place of V 2 ^i = etc. we obtain nV 2 ^i = piri, »Wa = H^, «V 2 fs = P^ and similarly ™V 2 %1 = PXU n V 2 %2 = P%2, WV 3 'X8 = P%3- (8) In place of the last stage which gave the metric criterion, we find , o ^ 2 %i j 2 d 2%2 ; o ^ \ oyoz dx^z d^ctyj J and finally, + ^^m-°- (9) These are the modified form of (5) and (7). 5". Inclusion of bodily forces, with particular reference to gravity on the surface of the Earth, and to centrifugal forces. In any case we shall have to consider the alteration made in equations (1), and their solution in (2) by the introduction 282 Mr. R. F. Gwyther on an Analytical Discrimination on the left-hand side of (1) of terms representing the com- ponents of the force per unit volume. These components can always be represented in a form similar to that given for the displacement in (3), but I shall suppose that the force per unit volume can be represented by the simpler forms B(F+/x) 5(F+/,) ^(F+/ 3 ) P ^5i~' p ^i~' p ~^~' and I shall suppose that we have selected the axes and that the yfrs are null. Then in (2) we must replace P by P + ^F+Z,), Q by Q + p(F+/,;, R by R + p(F+/ 8 ), leaving S, T, U unchanged. In the values found from the stress-strain relations there are no such changes to be made. We shall thus obtain from (6) V*0i-~tfi = V 2 2 - P f 2 = V 2 3 -pfi 2n{3 /5 F4/>(/i+/>+y8)} + (3m + n)V!(^i + ^+.ft) = 6m {'S? + a? + a?}- ' (10) If we write ^3 = </> + %3 + f + %/, where <£, % l9 ^ 2 , %3 have the values in (5) and (7) and may be regarded as Complementary Functions, then we remain with and 2n/) F + (m + n>(/ 2 +/ 2 +/ s ) + (m + n) V 2 <£' which may be regarded as giving the Particular Integral corresponding to the particular force acting. There are not many cases of interest. In the case of gravity on the surface of the Earth, as under natural forces generally, we have j\ =j 2 =fi = and (m + n) V 2 </>' + 2npF = 0. If we suppose ( — X, — ^, —v) to be the direction-cosines of the attraction of gravitation, F ^^gfrt + fiy + pz) and ^ = _^ (x ^ + , + v ^. of Elastic Stresses in an Isotropic Body. 283 and if we write P', Q', R' for the Particular Integral portion of P, Q, R — i. e. the terms which depend explicitly on g — we find 11X — it ?'=ffp\x + ~^gp(W + vz), . . . (12) with similar values for Q' and R', the Complementary Function part of P, Q, and R and the values of S, T, U being those given in the earlier part of this paper. The other case which I propose to consider is that of a body moving with angular velocities co x , co y , co z about the axes of coordinates which must be axes fixed in the body. It is implied either that the question is purely kinematica], or that a problem in Rigid Dynamics has been previously solved. The expressions for the acceleration of a point in the body are well known, and give for the effect of the reversed effective forces F = i{0)/ + »/> 2 + (w 2 + ft>*V+ (o> x 2 + a)/)* 2 — 2(o x (D y xy — 2co x o)zxz — 2co lJ (t) y yz } , with f l = x(yw,—z6)y), fz—y{z<d x —!cwz), f 3 = z(xw y -y6) z ). The form of the forces / x , / 2 , / 3 indicates that they will cause no strain in the body, and consequently cause no stress. If we proceed to find the effect which they have on the values of the stresses, they will be seen to disappear from the stress-equations. I shall therefore omit them for this purpose, and treat %/, yj > X$ as nu ^- We then find *' = ~ 12(Jt n) { (< + W * 2) * 4 + (ft>/ + "^ 4 (W/ + "^ - 2co y co z yz(y 2 + z 2 ) - 2co x co z xz(x 2 + z 2 ) — 2(D x (0 y xy(x 2 + y 2 )} s '=Hi' ete -' w with Complementary Function terms as before. These give no solution of any specific question. They only give a skeleton of the general form which a solution will take. and [ 284 ] XXI. Note on Damped Vibrations. By H. S. RoWELL *. IT is well known that the space time curve for free un- damped vibrations may be derived from the projection of a rotating vector, the end of which describes a circle, and it is fairly well known that for vibrations which are resisted by fluid friction proportional to the velocity, the space time curve may be projected (as remarked by P. G. Tait) from a rotating vector, the end of which describes an equi- augular or logarithmic spiral. The vibration of bodies when resisted by a constant frictional force — say solid friction — is of great importance in practical work and does not appear to have been adequately treated. The results obtainable are, moreover, in themselves of much interest. If F is the constant force of friction the equation of motion is m'x+c 2 x±¥ — 0, wherein the sign of F depends on the direction of motion. Substituting a? = X+ F/c 2 , we have F X = x + -j = A cos \\/m ' which gives a series of harmonic vibrations about alternating centres distant F/c 2 from the equilibrium position when * Communicated by the Author. Notices respecting Neiv Boohs. 285 friction is absent. The motion can be obtained by pro- jection from a spiral which is composed of semicircles as shown in the diagram. It will be seen that the amplitudes are in arithmetical progression and the difference for a complete period is 4F/c 2 , which may be called the arithmetic decrement. There is little purpose in using the ordinary definition of decrement, but it may be remarked that on this definition (i. e. ratio of successive amplitudes) the decrement ranges from unity for infinite amplitudes to infinity for zero amplitudes. The spiral curve described here does not appear to have been used before in scientific work, and it might be con- veniently called the arithmetic spiral or the spiral of semi- circles. XXII. Notices respecting New Boohs. Wmther Prediction by Numerical Process. By Lewis P. Richard- son, B.A., F.Inst.P. 4to, pp. xii + ^36. Cambridge Univer- sity Press. 30s. net. HPHE usual method employed in weather forecasting is a -*- development o£ that of Abercromby. Distributions of pressure are classified according to standard types, and the vari- ation on any occasion is predicted according to the behaviour of the atmosphere on previous occasions when conditions of the same type occurred. The method is therefore one of sampling inference in which the information utilized is all of one kind. Mr. Richardson believes that other information is relevant to the behaviour of the atmosphere ; and in this book he shows how to make use of the known results expressed in the hydrodynamical equations of motion and the equations of emission, transference, and absorption of heat and water. The method adopted is to work with equations each containing only one partial differential coefficient with regard to the time, so that this can be determined by means of the equation when the other quantities involved are known ; they include, of course, partial derivatives with regard to the position on the map and the height. These are to be found by observation at stations distributed according to a regular pattern, and the rate of change of each meteorological element at each station is to be calculated from them. Complications arise from the facts that the observations must be made at finite intervals both of time and of position, but these are allowed for. The stations required are more numerous than those at present in operation, and observations should be made every three hours to obtain the best results. Observations of upper-air winds and temperatures are required. 286 Geological Society : — The method is one that appeals strongly to the mathematical physicist. It is necessarily laborious in its present form, and probably could not be worked with sufficient speed to make it a practical method of forecasting ; but when forecasters have acquired experience in its use, they will probably find that a sufficient number of the quantities allowed for are comparatively small to make it possible to expedite the calculation considerably without great sacrifice of accuracy. The value of the work is not confined to the application to forecasting, though the possibility of predicting the disturbing occasions when cyclones cause merriment in the daily press by moving in the wrong direction makes this the feature of most general interest. Its discussion of the physical properties of the atmosphere is so thorough that it constitutes a text-book of the subject. Copious references to original literature are given, and any meteorologist requiring serious information on any topic will do well to look first in this book. The section on evaporation suggests that the only limitation on the evaporation from vege- tation is imposed by the difficulty of passing along the stomata tubes ; this is not always true even for an isolated leaf, and is certainly wrong for a carpet of grass, on account of the obstruc- tion offered by the vapour from one stoma to evaporation from another. The numerical data actually given, however, eliminate this source of error. Concerning the printing and style of the book, it is only necessary to say that it is published by the Cambridge University Press. The index is good. H. J. XXIII. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from vol. xliii. p. 1138.] February 1st, 1922.— Mr. E. D. Oldham, F.K.S., President, in the Chair. Mr. Cyril Edward Nowill Bromehead, B.A., F.G.S., de- livered a lecture on the Influence of Geology on the History of London. The 6-inch Geological Survey maps constructed by the Lecturer were exhibited, and some of the new features pointed out. The small streams now * buried ' are indicated on the maps, and the historical research involved in tracing them led to an appreciation of the connexion between the geology and topography on the one hand, and the original settlement and gradual growth of London on the other. The reasons for the first selection of the site have been dealt with by several writers : below London the wide allu- vial marshes formed an impassable obstacle ; traffic from the Influence of Geology on the History of London. 287 Continent came by the ports of Kent, and, if destined for the north or east of Britain, sought the lowest possible crossing of the Thames. This was near old London Bridge, where the low- level gravel on the south and the Middle Terrace deposits on the north approached close to the river-bank. A settlement was obviously required here, and the northern side was chosen as the higher ground. The gravels provided a dry healthy soil and an easily accessible water-supply ; they crowned twin hills separated by the deep valley of the Walbrook, bounded on the east by the low ground near the Tower and the Lea with its marshes, and on the west b}' the steep descent to the Fleet ; the site was, therefore, easily defensible. The river-face of the hills was naturally more abrupt than it is now, owing to the reclamation of ground from the river ; the most ancient embankment lay 60 feet north of the northern side of Thames Street. The first definite evidence of a permanent settlement was the reference in Tacitus. The early Roman encampment lay east of the Walbrook, and the brickearth on the west around St. Paul's was worked. Later the city expanded, until the St. Paul's hill was included, the wall being built in the second half of the 4th centuiy. The great Roman road from Kent (Watling Street) avoided London, and utilized the next ford upstream — at West- minster — on its way to Yeralamium and the north-west, The earliest Westminster was a Roman settlement beside the ford, built on a small island of gravel and sand between two mouths of the Tyburn. This settlement could not grow, as did London, since the area of the island, known to the Saxons as Thorney, was small. The road from London to the west joined the St. Alban's road at Hyde Park Corner, running along the ' Strand,' Avhere the gravel came close to the river ; a spring thrown out from this gravel by the London Clay was utilized for the Roman Bath in Strand Lane. Throughout Mediaeval times London was practically confined to the walled city, a defensible position being essential. The forests of the London-Clay belt on the north are indicated in Domesday Book and referred to by several writers, notably Fitzstephen, whose Chronicle also mentions many of the springs and wells and the marsh of Moorfields, produced largely by the damming of the Walbrook by the Wall. The same writer mentions that London and Westminster are ' connected by a suburb.' This was along the ' Strand,' and consisted first of great noblemen's houses facing the river and a row of cottages along the north side of the road ; this link grew northwards, at first slowly, but in the second half of the 17th century with great rapidity. By the end of that period the Avhole of the area covered by the Middle-Terrace Gravel was built over, but the northern margin of the gravel was also that of the town for 100 years, the London- Clay belt remaining unoccupied. The reason for this arrested development was that the gravel 288 Intelligence and Miscellaneous Articles. provided the water-supply. In early days the City was dependent on many wells sunk through the gravel, some of which were famous, such as Clerkenwell, Holywell, and St. Clement's. In the same way the outlying hamlets (for instance, Putney, Roehampton, Clapham, Brixton, Ealing, Acton, Paddington, Kensington, Islington, etc.) started on the gravel, but later outgrew it, as pointed out by Prestwich in his Presidential Address of 1873. In the City the supply soon became inadequate, or as Stow says ' decayed,' and sundry means were adopted to supplement it. The conduit system, bringing water in pipes from distant springs, began in 1236 ; London-Bridge Waterworks pumped water from the Thames by water-wheels from 1582 to 1817 ; the New River was constructed in 1613, and is still in use. It was not until the 19th century that steam-pumps and iron pipes made it possible for the clay area to be occupied, thus linking together the various hamlets that are now the Metropolitan Boroughs. Some of the ways in which Geology affects London to-day were briefly indicated, and the lecture was illustrated by a number of lantern-slides, reproduced mainly from old maps and prints. XXIV. Intelligence and Miscellaneous Articles. young's modulus and poisson's eatio for spruce. To the Editors of the Philosophical Magazine. Dear Sirs, — IN my recent paper in the Philosophical Magazine for May 1922 -*- there is an error on page 877. It is there stated that G"yz Gzy G"zx °~xz °~xy Gyx TP ' TH TH 5 XT* T? y ih z Jh z & x \h x Jbjy S are equal respectively to 700 S«, 700 S^ and ~ . £>x This should read E z _ Ojy E^ _ OjX Ea; _ Gjy Ey (TyS E z <T X z Ey (T yX S are equal respectively to 700 S«, 700 8 X and ~. The error becomes evident on reading the paper, but I xery much regret that it has crept in. Tours faithfully, The College of Technology, " H# CABBiwcwxMr. Manchester. May 24th, 1922. Tizahd & Pie. Corresponding to experiment A Fig. 2. Corresponding to experiment A In these photographs the lower horizontal line is the line of atmospheric pressure. The ordinates represent pressure, and the abscissae time. A is the beginning of compression, B the point of maximum compression, and C the explosion. The curve in the top left-hand corner is the cooling curve of the products of combustion. Phil. Mag. Ser. 6, Vol. 44. PI. FlG. 3. Corresponding- to experiment A Fig. 4. Corresponding to experiment Ds Fig. 5. Corresponding to experiment D-. THE LONDON, EDINBURGH, and DUBLIN: PHILOSOPHICAL MAGAZINE AND JOURNAL OF SO [SIXTH SEIU A UGUST 1922. XXV. On the Viscosity and Molecular Dimensions of Gaseous Carbon Oxysulphide (COS). By C. J. Smith. B.Sc, A.R.C.S., JD.I.C, Research Student, Imperial College of Science and Technology*. 'TPHE present research is a continuation of the work on JL the measurements of the viscosities of gases, for the purpose of elucidating the structure of the molecules constituting them. Some measure of success has attended this investigation in many cases where the necessary data are known, and suggests that an accumulation of further similar data may be fruitful. A case in point is that of the molecule of carbon oxysulphide, and this paper describes the measurements of the viscous properties of this substance. which is ordinarily gaseous. The data, hitherto unknown, which have been obtained, have been applied to calculate the molecular dimensions in the ordinary way. Apparatus and Method of Observation. The apparatus and method, which have been used to determine the viscosity of carbon oxysulphide, have recently been fully described f. Method of Experiment. The viscometer was carefully standardized with a new mercury pellet in the manner indicated in previous papers. * Communicated bv Prof. A. 0. Ranlrine, D.Sc. t A. 0. Rankine and C. J. Smith, Phil. Mao-. vo l. xlii. p. 60], Nov. 1921, and C. J. Smith, Proc. Phys. Soc. vol. xxxiv. p. 155, .June 1922. Phil. Mag. S. 6. Vol. 44." No. 260. Aug. 1922. U 290 Mr. C. J. Smith on the Viscosity and Molecular and the corrected time o£ fall proved to be 104*70 sees., a value which is probably correct to 0*1 sec. With this time of: fall the corresponding time of fall for carbon oxysulphide has been compared, and with appropriate corrections gives the relative viscosity, from which the absolute viscosity has been obtained by assuming the viscosity of air at 15 o, C. to be 1*799 x 10" 4 C.Gr.S. units. In addition, the variation of viscosity with temperature has been derived from com- parisons of the corrected times of fall at atmospheric and steam temperature. Preparation and Purification of the Carbon Oxysulphide. The carbon oxysulphide was prepared by the action of sulphuric acid (five vols, acid, four vols, water) on pure potassium thiocyanate in the cold (room temperature). At the same time hydrocyanic acid, formic acid, and carbon bisulphide are formed. To remove these impurities the method recommended by Moissan * was used. This consists In passing the gas through a strong solution of caustic potash to remove the hydrogen cyanide and then over wood charcoal to remove the carbon bisulphide. The gas was dried by being passed over calcium chloride, and then solidified at liquid air temperature. All permanent gases were pumped out of the U-tube containing the solid COS by means of a mercury pump. The liquid air was then replaced by a mixture of solid 00 2 and alcohol at —80° C, when it was observed that the vapour pressure of the liquid COS was about 30 cm. of mercury. The C0 2 mixture was then removed and samples of the gas collected over mercury. It was further purified before being introduced into the viscometer by fractional distillation at liquid air temperature. The liquid air having been replaced by C0 2 and alcohol at — 80° C, it was possible on account of the comparatively high vapour pressure of COS at this temperature to pump off successive quantities of dry COS sufficient io fill the viscometer at atmospheric pressure. Experimental Results. (Table I.) W^e have -f 15 = 69*96 sees., and £ 100 — 90*64 sees. The ratio of these times of fall gives the ratio of the viscosities at the corresponding temperatures. Thus *7ioo _ ^ioo _ 9 0*64 _-,. 9 QKp Vis ~ *i5 69 96 ~ * Moissan, Traite de Chimie, vol. ii. p. 318. Dimensions of Gaseous Carbon Oxysulphide. 291 Table I. Each time recorded is the mean of four observations in each direction for the whole pellet, and of three for the pellet wh divided into two segments. The letters in parent heses indicate t order in which the observations were made. en the Temp. (deg. C). Time of fall (sees.). Capillary correction (.r). Corrected time (0- Time at Whole pellet. Two segments. 15°0C. ioo°-oc. (a) 1529... 73-04 7635 00416) 7001 69-92 (6) 15-44... 73-08 76-25 00399 70-16 7005 — (e) 15-63 ... 7302 7622 0-0403 70-07 Mean 69-91 69-96 — (c)99-9 ... 91-51 92-43 0-0099 90-60 — 9063 (rf)99-9 ... 91-54 92-48 o-oioi 90 62 Mean — 90-65 9064 Assuming Sutherland's law to hold for this gas, the value of Sutherland's constant obtained is C = 330. Also at 15°'0 C, /cos = 69'96 4 ir 104*71) = 0-6682. Correcting for difference of slipping of COS and ah\, we obtain 259i= 0-6668. fail- Oil the assumption that the viscosity of air at 15°0 C. is 1-799 x 10- 4 C.G.S. units, the values for COS are Vl .= 1-200 x 10" 4 C.G.S. units and ?7 100 = 1*554 x 10" 4 C.G.S. units ; and by extrapolation using Sutherland's formula, ^=1-135x10- 'C.G.S. units. Calculation of Molecular Dimensions. The particular dimension calculated from the above results is the mean area A which the molecule presents in mutual collision with others. The basis of this calculation is Chapman's formula (Joe. cit.) modified in its interpretation in the manner suggested by Rankine. The value obtained is A = 106 x 10 _ir ' cm. 2 which may be subject to an error of 2 or 3 per cent. U2 292 -Prof. A. 0. Rankine on the Molecular Structure Summary of Results. Table II. Viscosity in C.G.S. units X 10-4. Sutherland's Mean collision area. em. a x'10— 1B . 15°0C. 100°'0 C. constant. o°-oc. i 1-200 1-554 1-135 330 1-06 In conclusion the author gladly acknowledges the grant for this research, which was made by the Government Grant Committee of the Royal Society, and also wishes to thank Professor Rankine for his continued help and advice. Imperial College of Science and Technology, S.W. 7. 1st May, 1922. XXVI. On the Molecular Structure of Carbon Oxy sulphide and Carbon Bisulphide. By A. 0. Rankine, D.Sc, Professor of Physics in the Imperial College of Science and Technology*. 1. FTHHERE are at the present day in the process of JL development several theories of atomic and mole- cular structure which are in many respects discordant. They have, however, at least one feature of general agree- ment — namely, the common view that the atoms of the inert gases occupy unique positions in the various schemes. The distribution of the electrons with respect to the nuclei in these atoms is regarded as having the characteristic of completeness, so that there is displayed no marked tendency to lose electrons or to capture additional ones. Moreover, atoms other than those mentioned are believed to have in varying degrees what may be called deficiencies and redundancies of extra-nuclear electrons, which they endeavour to adjust by entering into suitable combinations with one another ; so that either by the process of give and take, or by common use of the same electrons, con figurations corre- sponding closely to those of the inert atoms are attained bv the individual atoms forming the compound. 2. These views of chemical combination find their most * Communicated by the Author. of Carbon Oxysulphide and Carbon Bisulphide. 293 complete expression in the theory of Lewis unci Langmuir *, particularly in relation to the type of compound with which this paper is concerned — namely, that in which atoms, deficient in electrons, are regarded as sharing- them in order to reach the completeness of inert configurations. The main purpose of this paper is to apply the principles of this theory to the special case of the molecule of carbon oxysulphide, and to show that the molecular dimensions of this compound, as derived from viscosity data, are consistent with the Lewis- Langmuir view of its constitution. This test of the validity of the theory is made possible by the recent measurements by 0. J. Smith t of the viscous properties of the gas in question. Similar calculations for the molecule of carbon bisulphide have been made, and these await verification or otherwise when the necessary viscosity data are available. 3. Carbon oxysulphide belongs to a family of three compounds having the chemical constitutions C0 2 , COS, and CS 2 . The two former are gaseous at ordinary temperatures, and the latter a highly volatile liquid. In all of them carbon is a constituent, and COS can he regarded as the molecule obtained by the substitution of a sulphur atom for one of the oxygen atoms in C0 2 , or by the reverse substitution in CS 2 . It is probable that the carbon atom occupies the central position in each molecule, and that the nuclei of the three atoms lie in each case upon a straight line. 4. According to the Lewis-Langmuir theory (loc. cit.), the atoms in these molecules are linked together by sharing external electrons in such a manner that each atom approxi- mates to the configuration of the inert atom at the end of the corresponding row in the periodic table. Thus, in C0 2 the central carbon atom shares altogether eight electrons, four on each side with an oxygen atom. The electron configuration thus formed is that of three neon atoms in a row, for the inert atom corresponding to both carbon and oxygen is neon. In the molecule COS there are again eight electrons shared by the carbon atom, four on one side with the oxygen atom, and four on the other side with the sulphur atom. The electron arrangement thus attained is that of two neon atoms (corresponding to the oxygen and carbon) and one argon atom (corresponding to the sulphur). Applying a similar argument to the CS 2 molecule, we are led to regard it as resembling closely the electron distribution of inert atoms in the sequence argon-neon-argon in a line. In other words, we can treat each carbon or oxygen atom in a * I. Langmuir, Joura. Amer. Chem. Soc. vol. xli. p. 8C>8. t C. J. Smith, supra, p. 289. 294 Prof. A. 0. Rankine on the Molecular Structure molecule as having nearly the same dimensions as a neon atom, and each sulphur atom in combination as approxi- mating to the dimensions of an atom of argon. 5. The remaining question of how far apart are the nuclei of the atoms in the molecule finds a satisfactory answer in the work of W. L. Bragg*, whose X-ray -crystal measure- ments have enabled him to assign probable values for the radii of the outer electron shells of the atoms of the inert gases. The only values with which we are at the moment concerned are those of neon o and argon, which are given respectively as 0*65 and 1*03 Angstrom units. In cases like those under consideration, where outer electrons are playing a double part, the sharing is equivalent to contiguity of the outer shells, so that the distance apart of the nuclei is the sum of the radii of the appropriate inert atom shells. Thus for C0 2 , which is pictured as three neon atoms in line, the three nuclei are equally spaced and separated by distances equal to twice the radius of the neon outer shell, i. e. 2x0-65 A = 1'30 A. In COS the distance between the carbon and oxygen nuclei is the same, namely 1*30 A, but the distance between the carbon and sulphur nuclei is the sum of the radii of the outer electron shells of O Q O the neon and argon atoms, i. e. 0*65 A-f 1*03 A = 1*68 A. The three nuclei in COS are thus unequally spaced on account of the greater size of the argon atom. In CS 2 the distance between the o carbon nucleus and each sulphur nucleus is also 1*68 A, and the three nuclei are again spaced symmetrically. 6. It is evident that none of the three molecules under consideration, if their configurations are as indicated, can be expected to display spherical symmetry. In these circum- stances it is necessary to interpret in a special way the results of the well recognized method of calculating molecular dimensions from viscosity data. The quantity which is actually derivable from the formula is the mean value of the area which the molecule presents, for all possible orientations, as a target for mutual collision with other molecules in the gas. This area the author f has ventured to call the mean collision area, and its value for COS is given by C. J. Smith (loc. cit.) as 1*06 x lO" 15 cm. 2 The immediate problem before us is to find how nearly the tentative model of this molecule described above would exhibit this value for its mean collision area. The values of * W. L. Brag-o-, Phil. Mag. vol. xl. p. 169. | A. O. Kankine, Proc. Eov. Soc. A, vol. xcviii. p. 360, and Proc. Phys. Soc. vol. xxxiii. p. 362. of Carbon Oxy sulphide and Carbon Bisidplride. 295 the mean collision areas of the constituent configurations (which we are taking- to be those of neon and argon) are known, and it is usual to regard these symmetrical inert atoms as behaving as clastic spheres for purposes of collision. The radii of these collision spheres, as we may call them for the sake of precision, are 1*15 A and 1*44 A respectively, and they are considerably larger than those of the corre- sponding outer electron shells, so that they overlap when Fig. 1. — Molecular Dimensions from the point of view of the Kinetic Theory. The Carbon Dioxide Molecule: equivalent to three linked atoms of Neon, The Carbon Oxysulphide Molecule : equivalent to two Neon atoiris and one Argon atom linked together. The Carbon Bisulphide Molecule : equivalent to two Argon atom® linked together by one Neon atom. the nuclei are separated by the distances demanded by electron sharing. Fig. 1 shows three models, drawn to scale, representing what we may conceive C0 2 , COS, and 0S 2 to be like for purposes of intermolecular encounters. C0 2 may be regarded as three overlapping spheres,- each of the neon 296 Prof. A. 0. Rankine on the Molecular Structure collision size, with centres separated by the distances already •specified. In COS we take instead of one of the extreme neon spheres an argon collision sphere ; while in CS 2 both the outer spheres are of the argon size. In all three cases the diagram represents all the nuclei in th.Q plane o£ the paper, and the line joining them is evidently an axis of symmetry. 1£ these symmetrical axes are variously oriented, the area presented by the model assumes different values, and our problem is to calculate the mean value o£ this pro- jected area for comparison with that deduced from viscosity data. The author (loc. cit.) has already derived the necessary formulae for this purpose, and has shown that the result obtained by application to the first model in fig. 1, namely 00 2 , is very nearly equal to the actual mean collision area of the carbon dioxide molecule. In other words,, a carbon dioxide molecule behaves in collision as though it had the configuration of three neon atoms in a straight line and with outer electron shells contiguous. 7. Calculation for the COS Model. — In the model which we are taking to represent the COS molecule, the calculation in the strictest sense is greatly complicated by reason o£ the particular distribution of the spheres. The exact formulae which have been obtained {loc. cit.) for equal and unequal spheres only apply rigidly to cases where a special relation exists between the diameters of the spheres and the distances apart of their centres ; and the model under consideration does not fulfil this condition. But by regarding the problem from two different points of view, we can obtain, by means of the comparatively simple formulae already available, upper and lower limits which are so close together as to render unnecessary the laborious exact calculation. This course is all the more justifiable because it is fully recognized that the general treatment of the problem itself can only be taken as a first approximation to the truth. 8. Let us consider the effect on the area of projection of the model (reproduced in the full lines of fig. 2, a) caused by variations of orientation of the symmetrical axis joining the centres ]5 2 , and 3 of the constituent spheres. It will be convenient to speak of the sphere with centre L simply as sphere 1, and so on, and of the projections of the spheres, which will of course be circles, as projection 1 etc. As the axis 0, 3 approaches the line of sight, the projections of the centres approach one another, and the eclipsing of the spheres becomes more and more marked. Up to a certain point the total projected area is equal to the sum of Carbon Oay sulphide and Carbon Bisulphide, 297 o£ the areas of the whole of projection 3, the crescent formed by the overlapping of projection 3 over projection 2, and the crescent formed similarly by the eclipse of projection 1 by projection 2. Before the eclipsing of 2 by 3 is complete, however, projection 3 begins to encroach upon regions of projection 1 which are not already covered by projection 2. Tt is this fact that introduces into the exact treatment of the problem the complications to which reference has already been made. Thus in fig. 2, b, which shows the projected area for that orientation of the axis for which the eclipse of 2 by 3 is just complete, the crescent formed by 2 and 1 still Fiff. 2. •survives, but parts of it (as indicated by the shading) are covered by projection 3. The projected centres are 0/, 2 ', and (V respectively, and this particular state of affairs occurs when the angle between C^ 3 and the direction of projection is 9° 47' for the spheres having the dimensions and distribu- tion already specified. 9. Overlapping of the type just indicated, like all overlapping, has the effect of reducing the projected area ; it is therefore clear that if w r e neglect it we shall obtain too large a value for the mean area of projection — that is, an upper limit will be obtained by taking the mean collision area as the sum of the three parts : (a) the area of the circle 3, (It) the mean value of the area of the crescent formed by ^cireles 3 and 2 (c) the mean value of the area of the crescent 298 Prof. A. 0. Rankine on the Molecular Structure formed by circles 2 and 1. The first of these quantities is- the area of the central cross-section of the argon sphere itself, viz. 0*648 x 10 -15 cm. 2 ; the two latter are readily obtained from the graph in the paper already mentioned *.. They prove to be 0*217 x 10~ 15 cm. 2 and 0*226 x 10~ 15 cm. 2 respectively. The total is 1*09 x 10 ~ 15 cm. 2 , and this provides our upper limit. 10. With regard to the lower limit, we can obtain a satisfactory value by contemplating a variation of our model, which avoids the special type of overlapping responsible for complications. A suitable change for this purpose is to substitute for the sphere 1 a smaller sphere having the same centre but of such magnitude that its projection becomes just eclipsed by projection 2 at the same orientation of the symmetrical axis for which projection 2 is just eclipsed by projection 1, as shown by the dotted circles in fig. 2. o The radius of the necessary sphere is found to be 0*93 A as compared with the original value 1*15 A. Examination of the projection of a sphere of this size, in relation to the other two projections, shows that for no orientation does eclipsing of the shaded type appear, and the formulae already available enable the mean area of projection to be calculated exactly.. The value so obtained will, however, obviously be less than the true value aimed at, on account of the reduction of size assumed for sphere 1. Using the graph already mentioned,, the lower limit thus derived is 0*648 x 10" 15 cm. 2 -f 0*217 x 10~ 15 cm. 2 + 0*138 x 10" 15 cm. 2 = l-00xl0- 15 cin.* 11. The foregoing justifies the assertion that a molecular model having the dimensions of an argon atom succeeded by two neon atoms in line and spaced according to the- demands of outer electron contiguity may be expected to* have a mean collision area intermediate between 1*09 x 10- 15 cm. 2 and 1-00 xlO- 15 cm. 2 The actual value of the mean collision area of the COS- molecule, as determined from viscosity is with a possible error of 2 or 3 per cent. It falls definitely between the upper and lower limits obtained from our calculations, and seems to provide striking corroboration of * A. O. Rankine, Proc. Phys. Soc. vol. xxxiii. p. 371. of Carbon Oxysulphide and Carbon Bisulphide, 299 the theory upon which the estimates are based. But we must be content with the conservative remark that the dimensions of the carbon oxysulphide molecule, as found by the application of the kinetic theory, are consistent within the limits of experimental accuracy with the view that the three atoms of the molecule, by sharing external electrons, assume the electron configurations and behaviour in collision of particular groupings of the neighbouring inert atoms. 12. Calculation for the CS.> Model. — Although there exist at present no data for carbon bisulphide which suffice to calculate the mean collision area of the molecule in the gaseous state, the success of the previous comparison would appear to justify a prediction of its value by con- sideration of the appropriate model. This has been repro- duced in the full lines of fig. 3, a. Here again the model is one which does not lend itself to exact solution without laborious calculation ; but again, also, we can obtain satisfactorily close upper and lower limits. The area of projection will clearly be less than that corresponding to the model in which the dotted sphere is substituted for the small central one, so that we have three equal spheres of the argon size in line : it will, on the other hand, be greater than if the central sphere is entirely dispensed with, so that there are two equal argon spheres only, as represented in fig. 3, b. The dimensions of the spheres and the distances 300 Mr. F. P. Slater on the Rise of apart of their centres have already been stated ; and both modified models have mean areas of projection which are very easily calculated. The upper limit thus determined proves to be 2*12 times the collision area of the argon atom ; the lower limit is 1*90 times the same area. Using the known value 0*648 xlO -15 cm. 2 for the collision area of the argon atom, we find that the mean area of projection of the model consisting of two argon atoms with an intermediate neon lies between 1*37 x 10" 15 cm. 2 and l'23xl0- 15 cm. 2 We may venture to predict with some confidence that the mean collision area of the CS 2 molecule, when determined, will be found to be between the above values. A more exact estimate could of course be made, but the degree of accuracy at present attainable in determining molecular dimensions from viscosity measurements is not sufficient to render the additional calculation worth while. Summary. On the assumption of the validity of the Lewis-Langmuir view of molecular constitution, the probable behaviour during encounters has been examined for the molecules of carbon oxysulphide and carbon bisulphide. In the former case it is shown that the molecular dimensions as derived from the application of the kinetic theory to the viscosity measure- ments of C. J. Smith, are in striking accordance with the results of the above examination. In the latter case comparison is not yet possible, on account of the absence of necessary data. Imperial College of Science and Technology, May 11th, 1922. XXVII. The Rise of <y-Raij Activity of Radium Emanation. By F. P. Slater, MISc. (Vict.), B. A. [Cantab.)*. XN a previous paper t it has been shown how the initial rise of 7-ray activity, starting from pure radium emanation, depended on the nature of the walls of the tube containing the gas, the reason . being that a small but * Communicated by Prof. Sir E. Rutherford, F.R.S. t Slater, Phil. Mag. vol. xlii. p. 904 (1921). y-Ray Activity of Radium Emanation. 301 detectable y« radiation was excited in the walls by the impact of the a. particles emitted by the emanation. The amount of this excited radiation was, however, very small when the walls of the tube were composed of atoms of low atomic weight, and for a lining of pure paper the 7-ray activity of the emanation and its products was found to rise practically from zero. Under such conditions the 7 radiations from the tube are due only to the products radium B and radium C. Taking the number of emanation atoms disintegrating per second at initial time as unity, the number of radium-B atoms disintegrating per second at any subsequent time t is X 2 \ s 2 -Kit \ = 1,2,3 where \ 1? X 2 , A, 3 are the transformation constants of the emanation and the products A, B, and C respectively. This quantity is tabulated for various times up to 220 minutes at the end of this paper (Table II.). Similarly, the number of radium-C atoms disintegrating per second at time t is \= 1, 2.3,4 Tables for this quantity for various times up to 258 minutes have been given by Moselev and Makower * and by Rutherford t- The rise in 7-ray activity of a tube filled initially with pure emanation can therefore be represented by KXoAs^T e-M (A 2 -X 1 )(X 3 -X 1 ) \ =1,2,3 + (1-K)A 2 \ 3 \ 4 g e ~ Xl * 1,2, 3, 4 where K is the fraction of the ionization, measured under given absorption conditions, due to radium B when in radio- active equilibrium with radium C. Thus it is necessary to determine "K." Since the 7 rays •• Moselev and Makower, Phil. Mag. vol. xxiii. p. 302'(1912). t Rutherford, 'Radioactive Substances/ p. 490. 302 Mr. F. P. Slater on the Rise of from radium B are less penetrating than those from radium C, " K " depends on the thickness of matter through which the radiations pass before entering the ionization chamber. Rise curves have been experimentally determined for different thicknesses of absorption material, both lead and aluminium being used. The values of K for various thick- nesses have been deduced by trial, and are shown in fig. 1. Fig. 1, 0-30 V f - ;\ i i I > Mms. cf /ead. A comparison of the experimental and calculated rise curves of the 7~ray activity through 12'0 mm. of lead is given in fig. 2, After six minutes from the introduction of pure emanation, the calculated and experimental curves agree very closely. From these curves the absorption coefficient of the radium B-7 rays can be deduced, and the values found are given in Table I. along with comparative determinations by Makower and Moseley (loc. cit.) and Rutherford and Richardson *. The values of the absorption coefficients for the thick- nesses of aluminium are somewhat doubtful, since the supposition of homogeneity of the radium-C 7 rays is not justifiable through such small thicknesses. The increasing * Rutherford and Richardson, Phil. Mag. vol. xxv. p. 722 (1913). y-.Ray Activity of Radium Emanation. 303 value of fi (cm. -1 ) with decreasing thickness of absorption material (see Table I.) is to be expected, since Rutherford and Richardson (loc. cit.) showed that radium B emits certainly two types of radiation having absorption coefficients in aluminium of 0*51 cm, -1 and 40*0 cm. -1 , and possibly a third type (/a=230 , 0~ 1 in aluminium). Fig. 2; — liise of y activity from Radium emanation through ]2'6 mm. of lead. %Max. I f Activity T 1 C-30 I, Ex-perimem -a/ ri e / I; 0-50 /! fi 0-40 i f; 0-30 / / 0-20 /,' /> V 0-10 /,' ^/s' ■■'' ^' Y 77/77<P /o m/hs. The absorption coefficients in lead only, given in Table I., are corrected for obliquity of the rays entering the electro- scope, and King's correction is used as given in Case II. of his paper *, I* /O0- cos 0/O* sec 0] I 1 — cos where 1^ and I are the intensities of the radiation emerging through a plate of thickness t cm. and incident radiation respectively, /u, the absorption coefficient expressed in cm." 1 , and 6 the semi-angle of the cone of rays entering the electroscope. vol. xxiii. p. 248(1912). 304 Rise ofy-Ray Activity of Radium Emanation. Table 1. , Absorbing medium is Lead, except where otherwise shown. Thickness of Absorbing Plate. Value of jw (cm.— i) Moseley Rutherford for and and Radium-B rays. Makower. Richardson. 160-200 mm 2-7 cm.-l _ Varying from 100-150 ,, 2-8 cm -i .... 11-0 cm.-l 4-0- 6-0 „ 4-1 cm.- 1 4-0 cm.-l to (lead). 2-8 cm.-l in 1-5- 2-0 „ 6-2 cm.-l 6-0 cm.-l lead. 3-0- 4-0 „ (Aluminium). - 75 mm. (Aluminium) ... 1-7 cm.-l (Al) 10-0 cm.-l (Al) (lead). ? Table II. Rise of Radium B from Radium Emanation. Maximum = 0*97480 is taken as unity. Time in mins. 1 Calculated rise of Radium B. .. 0-00269 .. 0-01016 .. 002129 .. 003513 .. 005115 .. 0-06882 .. 0-08718 .. 010647 .. 0-12622 .. 0-14624 .. 0-16637 .. 0-18641 Time in mins. 14.. .. 16 18 20 30 Calculated rise of Radium B. ... 0-2260 ... 0-2649 ... 03023 ... 0-3379 . . . 0-4942 Time in mins. 110 Calculated rise of Radium B. ... 0-9502 2 120 1W .. 09643 3 ... 0-9750 4 140 150 160 170 180 190 200 210 214 220 ... 0-9835 ... 0-9886 6 7 8 40 50 <0 70 80 90 ..... 100 ... 0-6135 ... 0-7066 ... 0-7780 .. 0-8329 ... 0-8748 ... 0-9058 ... 0-9315 ... 0-9929 ... 0-9956 ... 0-9979 9 10 11 12 ... 0-9992 ... 0-9998 ... 0-9999 ... 1-0000 ... 0-9999 Summary. Curves showing the rise of 7-ray activity from pure radium emanation measured through a wide range of absorption thickness of matter have been determined and utilized in deducing the absorption coefficients of the hetero- geneous 7 radiation from radium B. My thanks are due to Professor Sir E. Rutherford for his invaluable help in carrying out this research, and to Mr. G. A. R. Crowe for the preparation of the radioactive material. [ 305 ] XXVI IT. An Experimental Test of Sm oluc ho iv ski's Theory of the Kinetics of the Process of Coagulation. By Jnanendra Nath Mukherjee, D.Sc, Professor of Physical Chemistry, University of Calcutta, and B. Constantine Papacon- STANTINOU, D.Sc, Assistant Professor of Chemistry, Uni- versity of Athens *. A short account of the Theory. IN some experiments on the degree of dispersion of colloidal arsenious sulphide on the rate of coagulation, it has been shown (J. Amer. (Jhem. Soc. vol. xxxvii. p. 2026, 1915 : and Sen, Trans. (Jhem. Soc. vol. cxv. pp. 467-8, 1919) that the finer sol is less stable. In 1915 one of us pointed out the obvious connexion with the increased facilities of coalescence. The smaller particles have a more vigorous Brownian movement due to the smaller frictional resistance of the medium. This would be clear from the well-known equation of Einstein. The diminution in the mean distance between the particles also increases the rate of collisions. It w r as stated that the adsorption theory does not take these factors into consideration. Recently Smoluchow T ski (Zeit. Phys. Chem. vol. xcii. p. 129, 1917) has been able to formulate the progress of the coalescence with time. His attention was drawn to the subject by Zsigmondy. Bredig (Anorganische Fermente, 1901, p. 15) suggested as the cause of coalescence an increase in surface tension with a decrease in the electric density on the particles. Zsigmondy (Zeitsch. Physikal. Chem. vol. xcii. p. 500, 1918) modified this idea in the sense that there is an attraction, between the particles which increases with decrease in the electric charge. As a result of this attraction he assumes that when one particle comes within a certain distance of another, the two coalesce. This distance is taken as a measure of the force of attraction and is called the radius of the sphere of action. It has been shown by Zsigmondy that the time required for a definite colour-change in a gold sol gradually decreases with rise in electrolyte concentration till it reaches a minimum t, which does not change any further with higher * Communicated by Prof. F. G. Donnan, F.R.S. t Similar minimum times have been observed with cupric sulphide and mercuric sulphide sols by the writers. A copper sulphide sol gave two minutes as the time necessary for the appearance of visible clots when the concentration of the precipitating electrolyte (barium chloride; was varied from N/300 to N/20. At dilutions higher than N/300 the time was observed to increase as usual (Mukherjee and Sen, loc. cit.). Phil. Mag. S. 6. Vol. 44. No. 260. Aug. 1922. X 306 Profs. J. N. Mukherjee and B. C. Papaconstantinou on concentrations. This was assumed to prove that the radius of attraction reached a maximum value. Smoluchowslu utilized this idea of a sphere of action to avoid a consideration of the forces that influence the coalescence. He considers the probability of particles coming within their mutual sphere of action when the radius of the sphere has a constant value determined by the conditions. It is assumed that as soon as a particle comes within the sphere of attraction by virtue of its Brownian movement the two particles coalesce. This dis- continuous view of the obviously continuous process of coalescence was assumed to avoid a consideration of the nature and distribution of the forces that are present. Considering the effect of the motion of each particle and also that each of the aggregates acts as a condensation centre, he derives the following equations : 2» = ^p , (l) ^=~°TV 2 , (2) K) (a.n .t) k l ". ^.(S^" • .... (3) where ' ; w " denotes the total number of particles originally present per unit volume before coalescence begins. They are all assumed to be spherical and equal in size. u t '•' is the time in seconds that has elapsed since the electrolyte and the sol have been mixed. " T " is a constant charac- teristic of the rate of coagulation and is given by T = ^Tr.D.Ka.V ^ where "D"* is the diffusion constant as given by Einstein's equation; a = 4.7r.D.Ra, and Ra is the radius of the sphere of action. TT 9 1 * 1) = ^ =77— — ■■- , where 11= the gas constant, 1 ° )7r • V •> S = the absolute temperature, N =Avogadro's number, »7 = the viscosity, and r= radius of the pavticle. the Kinetics of the Process of Coagulation. 307 %ti denotes the total number of particles in all stages of coalescence in unit volume when the time is " t " ; n l denotes the number of the primary particles whose original number was n Q at the time " t " ; n k denotes the number of particles of the ki\\ stage of coalescence — that is, the number of aggregates each of which consists of u k " of the primary particles. " k" is evidently an integer. In 1918 Zsig.nondy published the results of an investi- gation to test this theory. He restricted his investigation to the rate of decrease in the primary particles (green in the ultramicroscope) in a colloidal gold sol when the minimum time of coagulation has been reached. He found that Ra = 2*2 times r, the radius of the particles. Similar values were obtained by Westgren and Reitstotter (Zeitschr. Phijs. Chem. vol. xcii. p. 600, 1918) with more coarsely dispersed gold sols. The value of Ra/r, however, varied in one experiment from 1'4 to 3*8. The recent experiments of Kruvt and Van Arkel (Rec. Trav. Chim. Pays-Bas, vol. xxsisc. [4] p. 056, vol. xl. p. 169, 1920) show greater variations. They are of opinion that there is some regularity in these variations. They could not observe a maximum value of Ra/r equal to 2. They found a maximum value equal to 0'73. iSmoluchowski, assuming from the data of Zsigmondy available at that time that Ra/r = 2, points out that the maximum rate of coagulation is reached when each collision between two par tides is successful in bringing about a coalescence. When the rate of coagulation is slower, all the collisions are not successful in bringing about a coalescence of the particles. If " e " is the fraction of the collisions that are successful in bringing about coalescence, then " T ,J in equations (1) and (2) takes the form T _3 Np.iy r , 4Ra,rco.0.e' ' [ ° J where !N" , Ra, n , 6, and rj have the same meaning as in equations (1) and (2). Putting we nave ^^v__l (6) n n t l + /3.e.t (?) X 2 308 Profs. J. N. Mukherjee and 13. 0. Papaconstantinou on Since only " e " is variable, a comparison of the coagu- lation time " t " for the same change in the sol makes it possible to determine the variation in the percentage of successful collisions and its dependence on the conditions of experiment. When the maximum rate is reached, e = l and hence a measure of the absolute value of e is possible. Problems awaiting solution. — A. glance through the experi- mental work would show that the assumption of the constancy of "T" is not well justified. The simplicity of Smolu- cbowski's equations consist in that there is only one constant. The experimental limitations are great, and it is quite possible tbat the discrepancies are due to the defects of the ultra- microscopic method. The other possibility is that the simplifying assumptions of Smoluchowski — for example, the constancy of "T" independent of the stage of co^ alescence — are not true within narrow limits. Tt is of great interest to know the limits within which these equations are valid. The important questions that await solution in this connexion are : {a) the limits within which the above equations are valid ; and (b) if the above equations are valid, the variation of e with concentration of electrolyte ; (c) the dependence of e on the electric charge ; (d) the variation of e with temperature. In the following an account of an attempt to examine these factors, with the exception of (c) , is recorded. Indirect Methods. — Variations in physical properties that occur simultaneously with the process of coagulation can be utilized to measure the rate of coalescence. Smoluchowski pointed out that the viscosity measure- ments of Gann (Koll. Chem. Beihefte, vol.viii. p. 67 (1916)) do not satisfy the main requirements of his equations — namely, a similarity in the form of the curves (showing the variation in viscosity with time) independent of the nature of the electrolyte. He concludes that viscosity changes do not form a measure of the coagulation process. Yet he considers that the method is suitable for a quantitative comparison of the effect of various concentrations on the values of e when the curves are similar. The variation in physical properties, however, is likely the Kinetics of the Process of Coagulation. 309 to show the validity of the fundamental equations of Smoluchowski. The fact that the curves showing the change in viscosity with time are dissimilar shows that these assumptions are not justified, and Smoluchowski thinks that " T " is dependent on the magnitude of the aggregates. Since as yet it is not possible to express physical pro- perties — e. [/., the viscosity or the absorption of light — in terms of definite functions of the number and size of particles, a quantitative comparison of different sols is not possible by indirect methods. We have, therefore, to restrict ourselves to the same sol. Experiments icitJi Gold Sols. — An examination of the changes in the colour of gold sols on the addition of an electrolyte showed (Mukherjee and Papaconstantinou, Trans. Chetn. Soc. vol. cxvii. p. 1563 (1920)) that the variation in the absorption of light of gold sols affords an easy and accurate method suitable for this purpose. The gold sols prepared by the nucleus method of Zsigmondy conform very nearly to the requirements of equations (1) to (3) in so far as the particles are fairly uniform in size. It would be very convenient to work with a sol with re- producible properties, as data obtained on different dates with different preparations could be rigorously compared. It was found that a sol on standing for some time under- goes somewhat irregular changes, which may in part be due to dust particles getting in accidentally. In spite of all precautions, one cannot be sure that there is no such variation in a particular sample. This variation is not wholly due to the fungus that grows in these sols. For this reason it is necessary to vary one factor only at a time and compare its effects. The comparison was therefore restricted to the same sol so long as it showed no variation in its properties. The Constancy of^T" in Equations (1) to (3) during the Process of Coalescence. According to the simple assumptions of Smoluchowski, the progress of coalescence should be uniformly the same for various electrolytes and for their different concentrations. The constancy of u T " implies that if we assume a series of consecutive stages of coalescence of a sol — under a definite set of conditions, namely a definite electrolyte concentration 310 Profs. J. N. Mukherjee and B. C. Papaconstantinou on and temperature — following each other by intervals of time equal to "Si," they are each characterized by a definite number and manner of distribution of particles of each category (primary, secondary, etc.). Let us indicate the stage of coalescence corresponding to the time " t " seconds (since the sol and the electrolyte were mixed) under the given conditions by the numbers 2N, N x , N„ N ? „ . . . N* . . . , where the subscripts refer to the number of primary particles by the union of which the aggregate is composed. Thus Nfc denotes the number of aggregates, each of which is composed of "&" primary particles. "A:" is evidently an integer. Similarly let us denote the stage of coalescence corre- sponding to the time t' ( = t + At) by , 2N', N,', i\Y, N 3 ', ... K,/ ... . These stages of coalescence are independent of external conditions so long as equations (1) to (3) are valid. The only change that external conditions can bring about is a variation in the value of T — that is, if the external conditions are varied the sol will always pass through the same consecutive stages of coalescence and only the rapidity of succession of these stages will be determined by them. Any property which varies continuously with the progress of coalescence without having any maxima or minima can be utilized to characterize the stages of coalescence ; for each value of this property is characteristic of the time that has passed since the mixing of electrolyte and sol. According to the equations of Smoluchowski, the times taken to reach any particular stage depend only on the value of " T/' which is constant under a definite set of conditions. Let us compare two different electrolytes, A and B, of concentrations C\ and C 2 . Let us suppose that after the time " t " the stage of coalescence indicated by 2 N, N,, N 2 , N 3 , . . . N t . . . has been reached when the electrolyte is u A " of con- centration Ci. This stage of coalescence has a definite value for the physical property we are considering, and is independent of the value of T. Let us assum£ that " T l *'■ and '"T 2 " are the corresponding values of "T" for the two cases. To be definite, we shall consider the variation in the the Kinetics of the Process of Coagulation. 311 total number of particles of all categories, which varies continually with the progress of coalescence. Let us assume that at the times " ^ " and " £ 2 " both electrolytes have reached a state at which the total number of particles is the same. From equation (1) we have, therefore, l+rp 1 + rlT or t "~ r r * (9) The general equation (3) may be written as n (i + i) <\* +i (10) Since n and k are constants, if ^ is constant, n k has a fixed value — that is, the condition -J- =J- , which is deduced l-i 1-2 from the condition that ^n has a fixed value, also implies that the values of n u n 2 , ft3, ... n^ are the same in both cases. This means, in other words, that a definite value of %n fixes unequivocally the stage of coalescence. Therefore, from the deduction that the successive stages of coalescence are always the same and depend only on the time, any property of the sol that varies continuously can be utilized to re- present a fixed value of %n or n x or a definite stage of coalescence. A definite value of this property is thus characteristic of the stage of coalescence. It also follows from the above considerations that all curves showing a variation of this property with time should be similar. A deviation from this similarity, in itself, would mean that equations (1) to (3) do not represent the facts. The absorption coefficients of gold sols for different wave- lengths change on addition of an electrolyte in a complex manner. The theories of the colour of these sols as advanced by Maxwell Grarnett (Phil. Trans, vol. cciii. A, p. 385, 1904; vol. ccv. A, p. 237, 1906) and by Mie (Ann. der Phys. [4] vol. xxv. p. 377) would lead one to expect that any change in the number and manner of 3 1 2 Profs. J. N. Mukherjee and B. C. Papacon stun tin ou on distribution of the particles n 1? n 2 , etc. will produce a great change in the optical properties of the sol. This is in agreement with observations. Now, if the successive stages of coalescence were independent of the nature and concentration of the electrolyte, then the manner of variation of the complex absorption would be the same in each case. The absorption in the red region of the spectrum varies continuously, corresponding to each value of the absorption coefficient for a particular wave-length in this region ; the values in the other parts should be fixed. If the contrary holds good, then the conclusion is obvious that the successive stages of coalescence are not inde- pendent of the nature and concentration of the electrolyte as assumed by Smoluchowski. It has been found that for the stage indicated by the value of the coefficient of absorption for 683 /£/u = 0*4985, the values of the coefficient for the other wave-lengths given in the following table in column II. are independent of the nature of the electrolyte. The concentrations of the electrolytes were such as to produce rapid coagulation. In columns III. and IV. the coefficients of the "nucleus sol" have been given for the original sol and for the stage of coagulation cha- racterized by the value of the coefficient for 683 /xyu, = 0'4156 (Mukherjee and Papaconstantinou, loc. cit.). Wave-length, Table I. Absorption coefficients (Jc). iafifi. I. II. III. IV. 683 0-0453 0-4985 0*0376 0-4156 602 0-1055 0-3679 0-1131 0409 583 0-1518 0-3388 0-1595 0-3986 563 0-2076 0-3294 0-2076 0-336 547 0-2512 3238 0-2867 0-3732 523 0-3780 0-3780 0-3780 0-3882 506 0-4647 0-3581 0-3882 0-3780 475 0-3581 03198 0-3581 03780 Comparison of the Values of " T " as a Test of Smoluchowski' 's Theory. Since the absorption coefficient in the red region varies continuously with the coagulation and its magnitude is the kinetics of t lie Process of Coagulation. 313 sufficiently great, a definite value of the abso rption co- efficient for a fixed wave-length (683 /jl/jl) can be taken as representing a definite stage of the coalescence. In the following tables the absorption coefficients at different times are given for the wave-length 683 fi/ju. The tables are taken from the paper by Mukherjee and Papaconstantinou, loc. cit. Table II. Electrolyte : Potassium Chloride. Time in minutes, after mixing equal volumes of ( — electrolytes and sol. N/24. 0-0453 05 03732 1 0-438 1-5 0-4497 2 3 — 5 - 13 — 15 — Absorption coefficients for various concentrations N/26. N/28. 0-0453 0-0453 0-2867 01683 0-3630 02257 0-4046 — 0-438 0-2777 0-4497 0-3431 — 0-3836 — 0-4263 — 0-438 — 0-4497 Table III. Potassium Nitrate. Concentrations. Times. N/24. N/26. N/30. — 00453 00453 0-0453 0-5 0-3336 — — 1 0-4263 0-2866 0269 1-5 0-4497 0-3271 03143 2 0-3629 0-3336 3 0-4156 0-3732 4 0-438 0-394 5 0-4497 — 8 — — 0-4263 10 — — 0-438 16 — — 0-4497 314 Profs. J. N. Mukherjee and B. C. Papaconstantinou on Table I V. Barium Chloride. Concentrations. Times. * *• -. 0-852N/9OO. 0-852N/1000. 0-852N/1100. — 00453 0-0453 0-0453 1 0-2257 — 0-1603 2 0-2867 — 0-2007 4 0-3529 — 0-2687 5 0-3836 0-3051 0-3051 7 0-438 0-3431 0-3237 8 0-4497 0-3336 9 0-4497 03836 0-3529 11 — 0-4263 0-363 13 — 0-4497 0-363 16 — 0-3732 The limits within which the rate of coalescence could be varied were restricted by the fact that when the rate is slow the particles begin to settle, leaving a clear layer at the top, and the measurements are not reliable. Also, with time, some of the particles stick to the sides of the vessel. Lastly, it is difficult to avoid dust particles for a long time. The values given in Tables II.- V. were plotted graphi- cally, and the time intervals given in Tables V.-VI1. below were determined from these curves. Each of these curves is characterized by a definite value of T (or e). Corresponding to the three concentrations of any one of these electrolytes, there are three intervals which must pass in order that the absorption coefficient may have the same value. These intervals are co-related by the following relation according to equations (1) to (3) or (6) :— h _ ^2 % T, ~ T, ~ IV (11) 1 x 2 or ^:t 2 :^ 3 = T 1 .T 2 :T 3 = i:i:-. .... (12) e t e 2 e 3 Since T 1? T 2 , and T 3 are constant, the ratio of the time- parameters corresponding to the same absorption coefficient should be independent of the absolute value of the absorp- tion coefficient. Corresponding to different values of the absorption coefficient we get different values of t l9 t 2i and t 3 . All these values should show a constant ratio. In the the Kinetics of the Process of Coagulation. following three tables this comparison is made for the electrolytes mentioned in Tables II.— IV. Table V. 315 three Values of ibsorptiou coefficient. 0370 0-400 0-438 0-445 Absorption coefficients. 0-350 0-400 0-425 0-445 Absorption coefficients. 0-300 0-327 0350 0370 Electrolyte : Potassium Chloride Time in seconds. N/24. N/26. N/2S. t v t.,. *r 27 65 255 35-40 85 345 60 120 780 75 180 900 Ratios. Average Extreme deviation from average Table VI. Electrolyte : Potassium Nitrate. Tim<?s, -"-1 : T., : T,. : 2-47 : 9-44 : 23 : 93 2-0 : 13-0 : 2-4 : 12-0 1 : 2-3 : 11-4 7-4 % 19 % N/24. or. -i 1 07 30/^' N/26. 105 N/28. h. 145 1 Ratios, T 2 : : 39 : T 3 . 5-4 45 165 255 1 3-7 : 5 - 7 60 210 375 1 3-5 : 6-2 90 300 780 1 : 3-3 : 8-6 Average 1 : 36 : 6-3 sme variation from average 8-8% 33% Table VII. Electrolyte : Barium Chloride. Times. N/24. N/26. N/28. ** t,. k T i 135 277 345 1 165 345 430 1 225 430 540 1 265 480 780 1 Ratios T 2 2 2-1 1-91 1-81 Average 1 : 1'93 : 264 Extreme variation from average 6 % 13 % It will be seen from Tables V. to VII. that the agreement is as good as can be expected. The variation in T is as great as 11 times, but the ratios are constant. The agreement 316 Profs. J. N. Mukherjee and B.C. Papaconstantinou on shows that the ratios of the values of T are independent of the time or the stage of coalescence. The ultramicroscopic measurements so far made show even during one experiment a much greater variation in T, as will be evident from the following tables : — Table VIII. (a). (Observer : Zsigmondy.) Values of /3'= rTl . Series D. Series E. Series F, 0-083 0-105 0-040 0028 0-058 0-0195 0-0302 0-049 0-0183 00309 0-0475 0-0153 — 0-0403 0-0187 — — 00126 Zsigmondy used high concentrations of electrolyte for securing a rapid rate. When the rate of coagulation is slow and the duration of experiment is greater than a few minutes, he found that impossible values of /3' are obtained. He thinks that the presence of impurities in the water used in diluting the sol for ultramicroscopic observations is the cause of this irregularity. In his case ■the m iximuin time covered by the experiments is 80 sees. Similarly, Westgren and Reitstotter, working with coarse gold sols, find the following range of variation in the constant : — Table VIII, (6). (Observers : Westgren and Reitstotter.) Ra Values of — -* Series I. Series II. Series III. Series IV. 3-74 2-56 2-75 3-41 2-47 2-81 2-60 2-80 2-07 2-33 2-17 2-60 2-10 2-31 2-40 2-48 2-09 2-31 212 2-14 1-62 — — — 1-41 2-16 215 215 — 2'19 — 205 Average . 2-2 2-38 2-36 2-19 Extreme variation., .. 75% 10% 17% 55% the Kinetics of the Process of Coagulation. 317 Kruyt and Arkel *, working with selenium sol and very slow rate of coagulation, find extremely wide variations in T in the same experiment. Table IX. (Observers : Kruyt and Arkel.) Values o£ T (in hours) . I. II. III. IV. 2-8 200 131 1-3 51 390 55 34 44 270 52 2'2 (43) 320 54 43 (157) 600 68 105 200 - 370 55 40 — 510 — — — 440 48 37 — — — 52 __ _ — 38 The above few instances will suffice to show the range of variations in u T " during the course of one experiment that has been observed in the ultramicroscopic measurements. Considering that in Tables Y. to VII. the ratios between the different values of T are taken, the range of variation is extremely small. The actual deviations in the value of T in any one experiment must be much less than the extreme variations given. This comparison leaves no room for doubt that " T " is a constant in the case of gold sols and within the limits of the rate of coagulation that have been studied. In fact, these data constitute the best evidence so far recorded in favour of the theory of Smoluchowski. The Dependence of € on the Concentration. Tables V. to VII. show clearly how rapidly e, the percentage of successful collisions, increases with con- centration. A change of concentration in the ratio 24 to 28 increases the rate in the ratio 1 : 11 or 1:6 as the case may be. It would be extremely interesting to work with a sol which is less susceptible to impurities than these gold sols. * Fee. Trav. CLim. Pays-Bas, vol. xxxix. T4] p. 6o(i (1920) ; [4] vol. xl. p. 169(1921). 318 Profs. J. N. Mukberjee and B. 0. Papaconstantinou on Variation o/Tore with Temperature. Similarly, by determining the times required to produce a definite change in the colour o£ tbe sol for the same electrolyte concentration but different temperatures, we can determine the variation in e with temperature. From equation (3), we get 2 9 - = l+£V.«.« (13) Since a definite change of colour is being used, ^ is constant, or n 1 + • € f t = k^, a constant. , . . (11) Substituting the value of j3 in (14), we get 4 Ra . . n . ,^„ Since Ra, N , and n are constants, we have — — — = k' a constant (16) The viscosity of colloidal gold solutions has been found to be practically equal to that of water, and the variation with temperature can be assumed to be equal to that of water. For different temperatures Ave have £j . 0! . e l _ t 2 . 6. 2 . 6 2 - Vi V2 Since t x is experimentally determined and and 7) are known, variations in e can be compared. The experimental data are given below. They are taken from the same paper (pp. 1570-71). Table X. Temperatures. Electrolyte. Standards*. 15°. 30°. 50°. N/30 Potassium chloride .. V Sol. C. 5 inin. 10 -in in. 8 min. 30 sec. N/30 Potassium sulphate . ,, „ D. 30 sec. 10 sec. 10 sec. N/30 Potassium nitrate .. „ „ D. 42 ,. 18 „ 12 ,, * These i*efer to the protected gold sols used as standards for comparison of colour. See loc. cit. the Kinetics of the Process of Coagulation. 319 Table XI. Electrolyte: Barium Chloride. Sol. E. Con- centrations. Standards. Temperatures. 15°. 30°. 40°. 50°. 0-852 N/1000 v 2 7 min. 6 min 4 min. 50 sec. 4 min. 20 sec 0-852 N/1000 B 2 34 „ 23 „ — 13 „ 30 „ 0852 N/1200 V 2 23 „ 13 ., 12 min. 30 sec. 6 „ 15 „ 0-852 N/1200 B 2 124 „ 74 „ 62 min. — Con- Table XII. Electrolyte : Strontium Nitrate. Sol. F. Temperatures. centrations. Standards. 15°. 30°. 50°. N/1000 V 3 1 min. 10 sec. 20 sec. 8 sec. N/1000 B 3 8 „ 15 „ 1 min. 40 sec. 45 „ At 15°, 30°, 10°, and 50°, v /0 has the values 3"96 x 10~ 5 , 3-31 xlO- 5 , 2'1 x JO" 5 , and 1*7 x 10" 5 respectively. The values for the viscosity are taken from the tables in Kaye and Labv's hook on Physical and Chemical Constants, p. 30, 1919. From equation (17) we have 6 \o° " 6 30° : 6 -l(P ' 6 5U 3 = (v/tO)ro° : (v/tO'hoo : (r}/t6)^ : ( V /t0) 5O o. Table XIII. Temperatures. Electrolyte. N/30KC1 N//0X1O 7 N/SOK.SO^ N/30KNO 3 15°. 30°. 50°. 1-32 0-50 0-33 3-2 33 17 943 18-4 14-0 Table XIV. Electrolyte : Barium Chloride. Con- centrations. Standards. 0-852 N/1000 V 2 77/^ XlO 8 b; Ratio between f V., v/to 1 b; Tempera ti 0852 N/1200 V 2 rj/tOxW B 2 Ratio between J V a ,,-te [ B." 15°. 9 43 1-94 30°. 92 24 40°. 7-24 50°. 6-54 2-1 100 100 : 98 : 123 77 : 70 : 108 2-87 53-22 421 74-:> 2-8 04 45 100 100 : 148 : 140 : 98 : 106 : 158 320 The Kinetics of the Process of Coagulation. Table XV. Electrolyte : Strontium Nitrate. Temperatures. Concentrations. Standards. '77^ ^ 50°~^ N/1000 Y 3 77/^xlO 6 '565 1-65 2-1 B 3 ri/tBxlO 1 '80 3"3 377 Ratio between f V 3 100 : 291 : 371 nltB 1 B 3 ... 100 : 410 : 430 Since 7] = t6 is a constant for a definite electrolyte con- centration and temperature according to Smoluchowski's equation, the ratios should be independent of the standard used. This is true within the limits of experimental error with -852 N/1200 Barium Chloride. In the other two cases the variations are not great considering that we are com- paring the ratios. A slight variation in each value will be> magnified in the ratio. Taking into account the probable experimental error, it can be said that e is roughly constant in each experiment. On the other hand, the variation in e with temperature is considerable. We have already seen that the irregularity in the variation of e means that the precipitating power of the ions changes with the temperature (Mukherjee, Trans. Chem. Soc. vol. cxvii. p. 358, 1920). Further experiments with arsenious sulphide are in progress on similar lines. Summary. (1) It has been shown that the equations of Smoluchowski on the rate of coalescence of the particles of gold sols agree with the results obtained by the writers. (2) It has been suggested that the disagreement of the ultramicroscopic measurements with this theory may in part be due to the difficulties inherent in them. Our best thanks are due to Professor F. Gr. Donnan for his kind interest and encouragement, and also to our friend, Professor J. 0. Ghosh. Physical Chemistry Department, University College, London. [ bsi ] XXIX. The Adsorption of Jons. By Jnanendra Nath MuKHERJEE, D.Sc, Professor of Physical Chemistry in the University of Calcutta*. IN a paper in the Transactions of the Faraday Society (Far. Soc. Disc. Oct. 1921) an attempt has been made to define the nature of the adsorption of ions to which the origin and the neutralization of: the charge oE a colloidal particle are due. The origin of the charge was assumed to be due to the adsorption of ions by the atoms in the surface as a result of their chemical affinity. It was pointed out that the adsorption of one kind of ions will impart a charge to the surface, in virtue of which ions of opposite sign will be drawn near the surface. In the liquid there remains an equivalent amount of ions. of opposite sign. The electrical energy will be a minimum when these ious are held near the surface so that the distance between the oppositely charged ions has the minimum value possible under the conditions, and they will be held opposite to the ions chemically adsorbed. An "ion'' so held will not be " free" to move if its kinetic energy is less than " W " the energy required to separate the ion from the oppositely charged surface. The number of such "bound " ions deter- mines the diminution in the charge of the surface. When the concentration of ions of opposite charge in the liquid is small the number of ions "held" to the surface by electrical attraction will be small. If the chemically adsorbed ions have a valency equal to " N l5 " and if " N 2 " is the valency of the oppositely charged ions in the liquid in contact with the surface, then Ni.Xs.E 2 vv ~ D.a ' w where E = the electronic charge, x = the distance between the centres of the ions at the position of minimum distance, and " D " is the dielectric constant of water. Depending on the concentration of the oppositely charged ions in the liquid near the surface, at any instant a certain number of the " chemically adsorbed " ions are " covered " by ions of opposite charge. In the liquid near the surface there are always a number of free ions equivalent in amount to the " uncovered" chemically adsorbed ions on the surface. The total amount of ions of opposite sign both " bound" and * Communicated by Prof. F. G. Donnan, F.R.S. Phil Mag, Scr. 6. Vol. 44. No. 260. Aug. 1922. Y 322 Prof. J. N. Mukherjee on "free" is equivalent to the amount of ions "chemically, adsorbed." These " free" ions form the second sheet of the double layer. It is evident that as a result of their thermal motion the mean distance between the two layers will be greater than "#." The charge of the surface was treated as due to discrete charged particles widely separated from each other compared with molecular dimensions. It was shown in the previous paper that this view gives a rational explanation of the fact that a reversal of the charge of a surface can be brought out only by polyvalent ions of opposite charge. The equilibrium conditions were discussed and the equa- tions deduced were shown to be in agreement with the valency rule, the influence of the mobility of the oppositely charged ion, and with the influence of concentration on the charge o£ the surface. Only the theoretically simplest case has been discussed in the earlier paper. In the present paper the more important facts connected with the adsorption of ions are discussed from this point of view, and it will be seen that this view gives a simple explanation of most of the general conclusions already arrived at on experimental grounds. 1 heories of Adsorption. Before proceeding to discuss the adsorption of ions it will be convenient to deal briefly with the different views advanced to account for adsorption in general. The with- drawal -of a solute from a solution by a solid may be the result of the formation, of definite chemical compounds, of solid solutions, of mixed crystals and surface-condensation. In many cases all these changes are simultaneously present. In this paper the word " adsorption " denotes condensation or combination, at the surface only, without the interpenetra- tion of the adsorbed substance throughout the mass of the adsorbent (Mecklenburg's criterion, Z. Phys. Chem. Ixxxiii. p. 609 (1913) ; cp. also the sense in which the term is used in deriving Gibbs's equation). Faraday (Phil. Trans, cxiv. p. 55 (1834)) in his well-known explanation of the catalytic combination of hydrogen and oxygen on platinum surfaces,, remarks "that they are de- pendent upon the natural condition of gaseous elasticity combined with the exertion of that attractive force, possessed by many bodies, especially those which are solid, in an eminent degree, and probably belonging to all, by which they are drawn into association more or less close, without at the same time undergoing chemical combination though often the Adsorption of Ions. 323 assuming the condition of adhesion, and which occasionally leads under very favourable circumstances, as in the present instance, to the combination of bodies simultaneously sub- jected to this attraction." It is remarked further u that the sphere of action of particles extends beyond those other particles with which they are immediately and evidently in union, and in many cases produces effects rising into con- siderable importance.''' These remarks of Faraday mean, in modern terminology, that there is a sort of combination at the surface and that the transitional layer is more than one molecule thick. The subsequent views are in a way develop- ments of this conception. Gribbs treated adsorption from the standpoint of thermo- dynamics. A number of important investigations has been carried on by Milner (Phil. Mag. [6] xiii. p. 96 (1907)), Lewis (PhiL Mag. [6] xv. p. 506 (1908)), ibid. xvii. p. -466 (1909)), and Donnan and Barker (Prpc. Roy. Soc. Ixxxv. A. p. 552 (1911)). The present position is that the amount adsorbed is often considerably greater than what could be expected from Gibbs's equation. J. J. Thomson (' Applications of Dynamics to Physics and Chemistry ') showed that it follows from Laplace's theory of capillarity that in the surface layer between two liquids, chemical actions may take place which are absent in the bulk of the liquids. Lagergren (Biliang K. Svenska Vet. Hand. xxiv. p. 11, Xo. 115 (1898)) considers that adsorption in the surface of solids in contact with aqueous solutions is due to the com- pressed state of the water in the surface layer. On the experimental side the work of Freundlich and his collaborators — [Kapillar-Chemie, 1909 ; Z. Phys. Cheni. lix. p. 284 (1907); Ixvii. p. 538 (1909) ; Ixxiii. p. 399 (1910) ; Ixxxiii. p. 97 (1913) ; Ixxxv. p. 398 (1913) ; xc. p. 681 (1915) ; Koll.-Ch.em. Beihefte,\i. p. "297 (1914) : see also Schmidt, Z. Phys. Chem. lxxiv. p. 689 (1910) ; lxxvii. p. 641 (1911) : lxxviii. p. 667 (1912); Ixxxiii. p. 674 (1913); xci. p. 103 (1916). In the last-mentioned paper Schmidt and Hinteler conclude that Freundlieh ; s equation represents their experi- mental data better than that of Schmidt] — and of others, have shown that adsorption-equilibria can be generally expressed in terms of the well-known equation of Freundlich : xt)ll = a.C l P (2) Freundlich expressed the opinion that adsorption is mainly due to a decrease in surface tension as suggested by Gibbs, Y2 324 Prof. J. N. Mukherjee on In the case of adsorption of gases by solids, Arrhenius (Medd.f. k. Vef. Nobelinstitut, ii. N. 7 (1911); Theories of Solution, 1912, pp. 55-71) has drawn attention to the parallelism between the van der Waals's coefficient "a" for the different gases and the amounts of these gases adsorbed by charcoal, and he believes that this is definite evidence of the compressed state of the surface layer. At the same time he lays stress on the chemical aspect — namely, that in addition to the attractions between the molecules of the gas in the surface layer, one has to consider the chemical attraction of the surface atoms and the molecules of the gas. Recently, Williams (Proc. Roy. Soc. xcvi. A. p. 287 (1919) ; xcviii. A. p. 223 (1920); also Trans. Far. Soc. x. p. 155 (1914), in which complete references to the literature on negative adsorption are given) has treated adsorption from the points of view of Lagergren and of Arrhenius in a number of interesting communications. It may be mentioned here that the disagreement of ob- servations with calculations from Gibbs's equation is at least in part due to the fact that only one source of change in the free energy of the surface layer is taken into account. In the simplest case of the interface, liquid-saturated vapour (one component system), it is open to objection whether " 7 " denotes the total change in free energy of an isothermal and reversible-formation of unit surface. Bakker (Z. Phys. Chem. lxviii. p. 684 (1910)) has pointed out that if the density of the surface layer is different from that of the liquid in bulk a second term is necessary to represent the change in free energy. It is possible that in this particular case this second term is negligible in comparison with " 7," the tension per unit length at low temberatures, but at high temperatures " 7 ,7 has a low value and the saturation pressure is very great, so that the second term may be even more important. Williams (Proc. Roy. Soc. (Edinburgh), xxxviii. p. 23 (1917-18)) has drawn attention to the effect of the variation of the surface of an adsorbent when adsorbing — a factor which is very often neglected. Lewis (Z. Phys. Chem. lxxiii. p. 129 (1910) ; also Par- tington, c Text-book of Thermodynamics,' p. 473 (1913)) has discussed the influence of a variation in the electric density on the surface on the form of Gibbs's equation. These may be called the physical theories of adsorption. The difficulty in accepting them as general theories of ad- sorption is that they attempt to explain adsorption in terms of a single physical factor j e. g. diminution in surface energy the Adsorption of Ions. 325 or a layer under great internal pressure. The necessity for recognizing the existence of a sort of chemical interaction (as Arrhenius has suggested) becomes evident when one con- siders the specific nature of adsorption processes. This point has heen justly emphasized by Bancroft in recent years. Besides his papers in the ' Journal of Physical Chemistry,' compare 'Applied Colloid Chemistry/ 1921, p. 111). The chemical point of view has been put clearly by Lang- muir (J. Amer. Chem. Soc. xxxviii. p. 2221 (1916) ; xxxix. }). 18-18 (1917)). He believes that adsorption is due to the chemical affinities of the surface atoms. Considering the thermodynamic equilibrium between molecules of a gas at the surface and those in the surrounding gas he has deduced the following equations correlating the variation of the ad- sorbed amount with its pressure, where " di' is the fraction of the solid surface covered and is a measure of the amount adsorbed, vi is the rate at which the gas would evaporate if unit area of the surface were completely covered, "//," is the number of gas molecules striking unit area of the surface per second and is given by //,= 43*75 X 10~ 6 — — ;f- — — , and "p ,} denotes the pressure of the gas, "T" its absolute temperature, and " M" its molecular weight. a denotes the fraction of the total number of collisions of the molecules of the gas that leads to a condensa- tion on the surface; it is usually close to unity and evidently can never exceed unity. Some interesting applications of his theory to catalysis of gaseous reactions by solid surfaces are given, This theory explains many phenomena which are otherwise difficult to understand. Michaelis and Rona (Bio-Chem. Zeitsch. xcvii. pp. 56, 85 (1919)) conclude from the investigations of Michaelis and his co-workers that the assumption of special forces at the sur- face fails to account for the facts and that adsorption is the result of chemical affinity. I. The Adsorption of a Constituent Ion by a Precipitate. The adsorption of ions is different from the adsorption of neutral molecules or groups in that it introduces a new factor — an electrically charged surface. The variation in the electric charge enables us to follow the net effect of the adsorption of the two ions, as the electric charge depends 326 Prof. J. N, Mukherjee on only on the total number of ions (of both signs) fixed per unit area of the surface. Kataphoretic and electro-endosmotic experiments give us a quantitative idea of the relative ad- sorption of both ions. The electric charge helps to peptize the adsorbent, and a qualitative idea of the adsorption of ions can be formed from peptization by electrolytes. An insoluble precipitate formed by the union of two oppositely charged ions has a marked tendency to adsorb its component ions. In many cases the connexion between the adsorbed ion and the electrical charge has been established. These instances have been given in the earlier paper. The nature of the chemical forces responsible for this adsorption has also been defined. Instances of adsorption of ions as judged from peptization by electrolytes are given below. Bancroft (Rep.' Brit. Assoc, p. 2 (1918)) remarks: — ;; It seems to be a general rule that insoluble electrolytes adsorb their own ions markedly, consequently a soluble salt having one ion in common with a sparingly soluble electrolyte will tend to peptize the latter. Freshly precipitated silver halides are peptized by dilute silver nitrate or the corresponding- potassium halide, the silver and the halide ions being ad- sorbed strongly. Many oxides are peptized by their chlorides and nitrates, forming so-called basic salts. Sulphides are peptized by hydrogen sulphide The peptization of hydrous oxides by caustic alkali can be considered as a case of adsorption of a common ion or as the preferential adsorption of hydroxy lion. Hydrous chromic oxide gives an apparently clear green solution when treated with an excess of caustic potash ; but the green oxide can be filtered out completely by means of a collodion filter, a colourless solution passing through."" " Hanztsch considers that hydrous beryllium oxide is peptized by caustic alkali, copper oxide is peptized by con- centrated alkali, and so is cobalt oxide. In ammoniacal copper solutions part of the copper oxide is apparently colloidal and part is dissolved. Freshly precipitated zinc oxide is peptized by alkali, but the solution is very unstable " (cp. also negative hydroxide sols — Freundlich and Leonhardt, Koll. Chem. Beihefie, vii. p. 172 (1915)). At least in some of these cases the formation of new com- plex anions is possible, and it is not definitely known to what ion the peptization is due. Regarding the peptization of stannic acid gel by small quantities of alkali, Zsigmondy (Kolloidchemie, p. 122 et seq. (1920) ; also Varga, Koll. Chem. BeiJiefte, xi. p. 26 (1919)) remarks: "Dieses kann the Adsorption of Ions. 327 sowohl auf Adsorption des gebildeten Kaliumstannats wie audi daraut' zUruckt'iihren sein, das Kaliumhydrat mit den Oberflachenmolekeulen der Zinnsaureprimarteilchen in Reaktion tritt, wobei diese von der Oberflache der Primar- teilehen festgehalten werden." The view suggested bv the writer to account for the ad- sorption of a common ion, leads one to expect that ions which can displace one of the constituent ions in the crystal lattice should also be adsorbed. Marc {Z. Phys. Chem. lxxxi. p. 641 (1913)) has observed that crystalline adsorbents adsorb crystalloids to any marked degree only when they can form mixed crystals with them and are isomorphous with them. Paneth and Horrowitz (Physik. Zeitsch. xv. p. 924 (1914)) have noticed that oc the radio elements those only will be adsorbed that can form insoluble salts with the common ion of the adsorbent and can also form mixed crystals with the adsorbent. This kind of adsorption is somewhat different from the type we have considered for, as Paneth has pointed out in his case, an actual interpenetration of the two non- common ions is occurring in the crystal lattice. Thus radium is taken up by barium sulphate giving out to the solution barium ions in exchange. Such an interchange will not impart a charge to the surface. Attention may also be drawn to the explanation advanced by Bradford (Biochem. J. x. p. 169 (1916) ; xi. p. 14 (1917)) to account for zonal precipitations, first siudied by Liese- gang. Bradford thinks that the adsorption of a constituent ion is responsible for their formation. From the numerous instances given aboA r e, this conception seems to be quite plausible. It is probable that other factors have also an influence on the process (Hatschek, Brit. Assoc. Rep. p. 24 (1918)). II. The Variation of the Density of the Electric Charge with the Concentration of an Electrolyte. In the previous paper the particular case when the charge o£ the surface is due to strong chemical adsorption of ions of one kind and when the added electrolytes have not any ions,, subject to the cheniical affinity of the surface atoms, has been fully treated. In this case it was assumed that the number or! ions adsorbed at the surface by chemical affinity remains constant. The experimental data of Elissafoff on glass and quartz agree well with equations deduced from these assumptions, on the basis of the theory of electrical adsorption. 328 Prof. J. N. Mukherjee on The general case, however, is that : (a) At low concentrations the density of the charge on the surface at first increases to a maximum and at higher concentrations falls gradually towards a null value when the oppositely charged ions are monovalent. (b) On the other hand, when the oppositely charged ions are multivalent or complex organic ions the charge passes through a null value, becomes reversed in sign, and ugain reaches a second maximum, after which it falls slowly (Ellis, Z. Phys. Chem. lxxviii. p. &21 (1911) ; lxxx. p. 597 (1912) ; lxxxix. p. 115 (1914) ; Powis, Z. Phys. Chem. lxxxix., pp. 91, 179 (1914) ; Piety, Compt. Rend. cliv. pp. 1411, 1215 (1912); clvi. p. 1368 (1913); Young and Neal, J. Phys. Chem. xxi. p. 1 (1917); Krnyt, VersL Ron. Akad. v. Wetensch. Amsterdam, 27th Juin, 1914, also Koll- Zeitsch. xxii. p. 81 (1918)). The usual explanation is as follows: — The adsorption-isotherms for the two ions can be written as ^? =«_. A and ^ =«..«"*, ... (3) m K m A where the subscripts A and K refer to the anion and the cation respectively. To explain the increase in the charge at low concentrations it has to be assumed that a A >a K and fi A </3 K . .... (4) Thus in a paper read at the Discussion on Colloids arranged by the Faraday and the Physical Societies of London, Svedberg remarks : " Now as a rule, it happens that for the two ions of a salt both a and /3 have different values, e.g. a (cation) < a (anion) 8 (cation) >/3 (anion)." It is clear that the equation of the adsorption-isotherm can be reconciled with the first increase in the charge. But two objections can be raised against this empirical point of view. In the first place, no reason is given why the constants a and /3 shall have generally the relative values assumed above for the cation and the anion. Secondly, these assumptions can- not explain the second maximum charge and the subsequent decrease observed with multivalent ions of opposite charge. It will now be necessary to assume that « A <« K and /3a>/3k, .... (5) in direct contradiction to the assumptions already made the Adsorption of Jons. 329 (r/>. (4)). Besides, one cannot get any idea lis to why the anion is generally more strongly adsorbed at low con- centrations. The facts can, however, be explained as follows : — The negative charge of surfaces in contact with water is to be sought for in the chemical natures of the anions and the cations. The simpler electrolytes (excluding dyes and complex organic ions) have cations whose chemical behaviour can be referred simply to the tendency of the component atom (e g. of the alkali and alkaline earth metals) to pass into the ionic state. These ions do not form any complex ions. They form only one type of compounds that are stable in aqueous solutions, namely, electrolytes with the atom existing as a positively charged ion through the loss of one or more electrons. On the other hand, the anions in general form types of compounds other than electrolytes, and also form complex ions. It is, therefore, possible to imagine that anions are subject to the chemical affinity of the surface atoms and that the chemical action on the cations is relatively small. Complex cations like those of the basic dyes should, for the same reason, be easily adsorbable. This is a well- known fact. If now, the assumption is made that the chemical affinity acting on the anion of the electrolyte added is stronger than the electrostatic attraction of the surface on the cation, the observed variation of the charge with the concentration of the electrolyte is easily accounted for. This case corresponds to a strongly marked maximum of a negative charge at a low concentration of the electrolyte. The initial charge of a surface in contact with pure water -can be due either : (a) to the strong adsorption of an ion of a minute quantity of suitable electrolyte associated with the solid, (7>) or to the adsorption of hydroxy! ions from water. On the addition of an electrolyte the density of the electric charge will increase at low concentrations because of the chemical adsorption of the anion. The electrical adsorption of the cation is smaller as the chemical adsorption has been assumed to be stronger. Besides, the electric charge of the surface is also not at its maximum. As the surface becomes more and more covered by the anions the rate of adsorption das/dc — where " da" is the increase in the amount adsorbed per unit surface due to an increase in the concentration ^ dc" — rapidly decreases. Also, the electric charge repels the anions, and those only can strike on it that have sufficient 330 Prof. J. N. Mukherjee on kinetic energy to overcome the potential of the double layer. The number of collisions is thus not proportional to the concentration but rises more slowly. Near about the point where the surface becomes saturated the value of dxjdc will be almost zero (cp. the shape of the adsorption-isotherms of Freundlich, Arrhenius, and Langmuir). On the other hand, the electrical adsorption increases continually with the concentration and the increase of the charge. It is- apparent that soon a balance will be reached between the chemical adsorption of the anion and the electrical adsorption of the cation. The minimum charge will correspond to the stage when dxjdc for the cation is just equal to dxjdc for the- anion. Beyond this concentration the charge will decrease rapidly,, and when the surface has been saturated with the anion the subsequent variation in the charge is simply due to electrical adsorption. The reversal of the charge by electrical adsorp- tion has been discussed in the earlier paper. It is necessary to add that as the electrically adsorbed polyvalent cations impart a positive charge to the surface, the adsorption of the cation decreases and the electrical adsorption of the anion becomes possible. As long as there is a positively charged surface the adsorption of the anion will increase more rapidly with the concentration than that of the cation. A second maximum will thus be reached and a decrease in the charge will follow. The electrical adsorption of the anion is small because of the smallness of the positive charge and an initially existing negatively charged surface. A further reversal of the charge is not possible, and, in fact, has never been observed. III. The Actio?i of Acids and Alkalies. The works of Perrin and of others (J. Chim. Phys. ii. p. 601 (1904) ; iii. p. 50 (1905) ; Haber and Klemensie- wicz, Z. Phys. Chem. Ixvii. p. 385 (1909) ; Cameron and Oettinger, Phil. Mag. [vi.] xviii. p. 586 (1909)) have shown that hydrogen and hydroxyl ions behave exceptionally in that they impart to the surface a charge of the same sign as they carry. This behaviour is in contrast to that of the other univalent ions. Perrin attributes their singular activity to the smallness of their radii. In order to explain the presence of these ions, in excess, in the surface layer, it is necessary to assume some sort of a restraining force acting on them at the surface. Haber and Klemensiewicz consider that there is an ad- sorbed layer of water in the surface by virtue of which the- the Adsorption of Ions. 331 solid .acts as a sort of combined hydrogen and oxygen electrode. They treat the subject from the points of view of thermodynamics and Nernst/s theory of electrolytic solution tension. It has been pointed out by Freundlich (and Elissafoff, Z. Phys. Chan. Ixxix. p. '407 (1912)) that hydrogen and hydroxyl ions are not the only ions which impart a charge to the surface. In many cases, acids haye been observed not to reyerse the charge at all. Many sub- stances haye a negative charge in contact with pure water. These facts show that selectiye adsorption of hydroxyl ions has also to be considered. This thermodynamic treatment from the point of yiew of Nernst's theory does not attempt to explain electro-endosmosis. For this purpose it is necessary to conceive of an electrical double layer, of which the layer imparting a charge to the surface is fixed relative to the mobile second layer. Freundlich, and Freundlich and Rona (Koll. Zeit. xxviii. 5, p. 240 (1921); Kgl. Preuss. Akad. Wiss. Berlin, 1920, p. 397, C. 1920, iii. p. 26) have shown that the potential measurements by Haber's method are not in agreement with those measured by electro-endosmotic experiments. They therefore suggest that there are two distinct drops in potential as one passes from the solid to the liquid (glass to water). The first drop is wholly in the solid and is probably of the nature associaied with the Nernst theory of electrolytic solution-tensions. The second drop is in the liquid and composes the Helm- holtzian double layer which it is necessary to assume to explain electro-osmosis and cataphoresis. At the same time the characteristic effects of hydrogen and hydroxyl ions on neutral substances like barium sulphate, silver chloride, naphthalene, etc., point strongly to the correctness of Haber's fundamental assumption that the explanation is to be sought in the equilibrium between the hydrogen and hydroxyl ions in the adsorbed layer of water and those in the bulk of the liquid. Williams (Proc. Roy. Soc. xcviii. A. <p. 223 (1920)) has recently suggested that the layer of water adsorbed on a charcoal surface is under great internal pressure (about 10,000 atmospheres). Applying Planck's equation he shows t}iat the effect of this pressure will be to increase the con- centration of hydrogen and hydroxyl ions in this layer. This increased concentration will setup a diffusion potential. He draws attention to the difficulties in accepting this view of the origin of the potential difference at the surface. In the instances considered by Haber and Perrin, the solid has 332 Prof. J. N. Mukherjee on little or no potential difference in contact with pure water, and the considerations developed by Williams are not applicable. Case 1. — The surface is inert. We shall assume that the atoms in the surface do not exert any chemical affinity on hydrogen and hydroxyl ions as such, or on the dissolved acid (or alkali) with which it may be in contact. The adsorbed water molecules behave as a solid layer, being held by strong chemical forces (Haber, loc. cit. ; Hardy, Proc. Roy. Soc. lxxxiv. B. p. 217 (1911)). It is clear that the surface will be neutral in contact with pure water. The molecules of water in the adsorbed layer are in thermodynamic equilibrium with those in the bulk of the liquid. It is reasonable to imagine that a transfer of an electron is taking place between the hydrogen atom and the hydroxyl group in the water molecules in the surface layer, as it does in the molecules in the liquid. That is, the water molecules are dissociating into ions at a definite rate. Let ki nx" be the number of water molecules (in the adsorbed layer) passing into the ionized phase per unit area per second. For equilibrium, as many hydrogen and hydroxyl ions are uniting to form neutral water molecules. Since the adsorbed water molecules behave as a solid layer, recombina- tions would take place mostly between adjacent hydrogen and hydroxyl ions. The recombination will be extremely rapid. It can be assumed that at any instant the number of hydrogen or hydroxyl ions actually remaining free in the surface will be a negligible fraction of the total number of water molecules. The neutralization of the ions being formed in the surface layer can also be brought about by impinging hydrogen or hydroxyl ions present in the liquid. In contact with pure water the probability of such collisions is small, for the concentration of hydrogen and hydroxyl ions is extremely small. Thus neutralization of the ions being formed in the surface layer is possible in two ways : (1) H s ° + OH, 1 — ^HOH— the subscript "s" refers to ions in the surface layer ; (2) (a) H.o + OH/— ->HOH, (6) H/>+ OH, 1 — *HOH— the subscript «f» refers to the freely moving ions in the liquid. In contact with pure water, neutralizations according to scheme 2 are small in number. Also 2 (a) and 2 (b) are equally probable. Consequently the numbers of H s ° and OH, 1 remaining in the surface at any instant will be equal, and the surface will be neutral. the Adsorption of Ions. 333 When an acid is added to the water the neutralizations according to scheme 2 (a) will he completely negligible, but those according to scheme 2(b) will not be so. The total number of neutral molecules of water formed in the surface is still equal to " .r," but a number of them is now being formed according to 2 (/>). Corresponding to the number of neutralizations according to 2 (6), a number of hydrogen ions will remain in the surface layer in excess of the number of hydroxyl ions. The rate at which 2 (/>) proceeds thus determines the free charge on the surface. An equivalent number of anions remain unneutralized in the liquid and form the second mobile sheet of the double layer. The free charge on the surface will evidently increase with rise in the concentration of hydrogen ions in the solution. There are, however, two factors opposing this increase in the charge of the surface. A. The proportion of hydrogen ions striking on the surface diminishes as the positive charge of the surface increases. Only those ions which have sufficient kinetic energy to over- come the electrical repulsion can reach it. If e be the potential of the double layer in C.Gr.S. units, then the number of collisions of the ions per unit surface per second is pro- portional to U H o.C E o.^- E /^ ..... (6) where H o denotes the concentration of free hydrogen ions in the liquid, "E" is the electronic charge in C.Gr.S. units, T is the absolute temperature, Uho is the mobility of the hydrogen ions in water, and K = E/No, where " R " is the gas constant and No the Avogadro number. B. The other factor that tends to diminish the charge of the surface is the electrical adsorption of the anion of the acid added to the solution. That this plays an important part will be evident from the following examples taken from the observations of Perrin : — Rate of Electro- Substance. Electrolyte. endosraotic outflow. XUOr, M/1000 HC1 +110 „ M/1000 citric acid + 5 ,, M/1000 HNO :i (or HOI) +100 „ M/1000 H 2 S0 4 + IS OCl 3 M/1000 IIN0 3 + 85 „ M/1000 H.^SO, + 21 ,. M/500HO1 + 90 , M/1000 H, C 2 , + 30 „ Feebly acid with HC1 + 75 ., Solution of KH, (P0 4 )1 with approximately the same number of free hydrogen ions as above 334 Prof. J. N. Mukherjee on Both these factors tend to diminish the rate of increase of the charge with rise in the concentration of hydrogen ions. For acids with simple univalent anions, the electrical ad- sorption at low concentrations can be left out of account in view of the excessive mobilty of the hydrogen ions. A quantitative relationship can now be obtained between the charge on the surface and the concentration of the acid. Let x 1 be the rate of neutralization according to 2 (6) above. We have then ^K.^.Cho.-*- 3 ^, .... (7) where k is a constant. The density of the charge on the surface is proportional to x 1 — which is a measure of the number of hydrogen ions remaining in excess in the surface. If the thickness of the double layer remains constant then the potential of the double layer is proportional to the density of the charge : that is, to x 1 . When all the hydroxy 1 ions in the surface layer are being- neutralized according to 2 (b) the surface will have a maximum charge determined by i( x." Putting x\/% = 0, since u x" is a constant, we have e*6, (8) and 6 represents the ratio of the hydrogen ions present in excess at any instant in the surface layer to the maximum number possible when the neutralization takes place only according to 2 (b). The potential of the double layer can be written as e = h . x 1 = k 2 . 6 = h . Oho . e-*- ^ . U H o, . . (9) or # = £ .Cho.^-*/ t .U H o, (10) where k\, Ic 2 , fe, Ic and /3 denote constants. Similarly, for alkali solutions we have e=ko.Gom.e-^^.UoKi. ...... (11) The maximum charge, being determined by a, will be the same with alkali as with acid. Of course, the influence of the oppositely charged ion in the acid or the base is being- neglected. Case 2. — 2 he surface is not chemically inert : preferential adsorption of one ion is possible. A review of the literature shows that surfaces in contact with water are seldom neutral. They are generally more err less negatively charged. This is intelligible in view of the the Adsorption of Ions. 335 chemical reactivity of the hydroxyl group. The presence of the potentially tetravalent oxygen atom possibly leads to a selective adsorption of hydroxyl ions by most surfaces. Thus glass and quartz have a marked negative charge in contact with water (cp. Elissafoff). On the addition of an acid the electrostatic forces will produce a diminution of the charge. The electrical adsorption of hydrogen ions by hydroxyl ions cannot be distinguished from the recombination of hydrogen and hydroxyl ions to form neutral molecules of water. This is confirmed bv the fact that the equation of electrical ad- sorption (cp, previous paper) satisfactorily represents the diminution of the charge. Perrin [loc.cit) found that, excepting alumina and chromium chloride, all other substances (naphthalene, silver chloride, boric acid, sulphur, salol, carborundum, gelatine, and cellulose) show a preferential adsorption of hydroxyl ions. The sur- faces have a negative charge even in contact with acid solutions. He also found that at higher concentrations of the acid the surface acquired a positive charge. Elissafoff, McTaggart, Ellis, Powis, and others could not observe this reversal in their investigations. Electrical adsorption of hvdrooen ions cannot lead to a reversal of the charge. The reversal (or the non-reversal) of the charge becomes intelli- gible if it is assumed that the considerations set forth in deducing equations (8) or (9) are correct. In contact with pure water the surface has a layer of adsorbed water and a number of hydroxyl ions. The amount of hydroxyl ions adsorbed by the surface will, in general, be small, as the concentration of the hydroxyl ions is very small in pure water. If, however, the adsorption is very strong the surface will have a considerable negative charge. On the addition of an alkali the negative charge of the surface will increase, due to two reasons: (1) the preferential adsorption of hydroxyl ions will increase, and (2) the number of hydrogen ions being formed at the surface will be more and more neutralized by hydroxyl ions in the liquid (cp, scheme 2 {a)). A maximum will be reached when the surface is saturated by preferential ad- sorption and when — \ in equation (9). The maximum charge per unit area can be written as E ro =# + y (for alkali), .... (12) where " os" corresponds to the charge when = 1 in equation (8) and " y" is proportional to the number of 336 Prof. J. N. Mukherjee on hydroxyl ions the surface can adsorb per unit area when it is saturated. Since the chemical adsorption of hydrogen ions is assumed to- be absent, on the addition of an acid the negative charge will decrease owing to electrical adsorption till the surface becomes neutral. At this concentration of the acid, the surface has an adsorbed layer of water, and an equal number of hydrogen and hydroxyl ions, An increase in the positive charge cannot be due to electrical adsorption of the univalent hydrogen ions (cp. previous paper). The increase in the charge is due to the neutralization of the hydroxyl ions being formed in the surface by impinging hydrogen ions, as represented in scheme 2 (/>) above. The maximum charge E 9rt for an acid will, therefore, be equal to " x, 9i 2 he maximum charge due to acids thus gives a measure of the hydration of the surface. The difference between the maximum charge observed with acid and with alkali gives a measure of the amount of hydroxyl ions that is required to saturate the surface. In trie preceding discussion, the chemical and electrical adsorption of the anion of the acid lias been left out of account for the sake of simplicity. If the initial negative charge of the surface in contact with pure water is consider- able the electrical adsorption can be complete only at high concentrations of the acid, i. <?., the surface will be neutral at a high concentration of the acid. The electrical adsorption of the anion, is no longer negligible. A reversal of the charge, though theoretically possible, may not be actually observed owing to the great concentration of the anion. The reversal is thus dependent on : — (1) a. large value of x, and (2) a small value of y. A non-reversal is to be expected when the opposite is the case, ?'. <?., (1) a small value of " x" and (2) a large value of "?/•" A regular transition from marked reversal to non-reversal can be observed in Ferrin's work. With cellulose he also does not record a reversal of the charge. It is to be expected from the preceding considerations that non-reversal will not be observed when the concentration of the acid required to render the surface neutral is comparatively high, i. <?., the anion concentration is high. The concentration of the acid in the case of cellulose is the greatest recorded by Perrin. the Adsorption of Ions. 337 The chemical adsorption of the anion is also not to be neglected. The experimental data on this subject are meagre. The various points raised here can be experimentally eluci- dated. As shown above, the standpoint developed in this paper can correlate all the observed tacts. Besides, it gives- a definite idea of the electrical double layer. Adsorption of electrolytes. In the preceding sections the adsorption of ions has been considered with reference to the electrical charge of surfaces- in contact with aqueous solutions of a single electrolyte. The electric effects accompanying the adsorption of ions have enabled us to follow the total adsorption of ions of both signs. In considering the adsorption of ions measured by chemical means it is important to remember the influence of the ad- sorption of the solvent pointed out by Arrhenius, Bancroft, Williams, and others. The amount adsorbed is small and the analytical measure- ment is difficult. For this reason, investigations have centred round adsorbents with great adsorbing power and substances which are strongly adsorbed. Often it happens, that if a sub- stance is used in a satisfactorily pure state it does not have the necessary specific surface to make the estimation of the adsorbed amount possible. As a result adsorbents generally contain small amounts of other substances. The importance- of these impurities has been pointed out by some investigators. Michaelis and Freundlich and their co-workers have done systematic work in this field. Their investigations have brought out the following regularities : — («) The electric charge of the solid influences the ad- sorption, Thus Michaelis and Lachs (Z. Elektro-Cliem. xviL pp. 1, 917 (1911)) ; Biochem. Zeitsch. xxv. p. 359 (1910) ) x and Davidsohn, Biochem. Zeitsch. liv. p. 323 (1913)) found that in contact with acid solutions chnrcoal adsorbs anions strongly and does not adsorb cations. The reverse happens in the case of cations. Freundlich and Poser [Koll. Cliem.. Beihefte, vi. p. 297 (1914)) undertook an extensive investiga- tion, and they agree with Michaelis as to the electro-chemical nature of the adsorption. (l>) The chemical nature of the adsorbent has a specific action. Michaelis and Rona {Biochem. Zeitsch. xcvii. pp. 57, 85 (1919)) believe that adsorption is due to chemical affinity. They mention that charcoal has a great capacity for adsorbing substances containing a chain of carbon atoms. (Cp. Abder- halden and Fodor, Fermentforschuncj, ii. p. 74 (1917).) Phil. Mag. S. 6. Vol. 44. No. 260. Aug. 1922. ^ Z 338 Prof. J. N. Mukherjee on Freundlich and Poser (loc. cit.) found that the nature of the adsorbent plays an important part in determining the ad- sorbability of a dye. Both Michaelis and Freundlich agree that at least two types of adsorption of ions can be recognized. (c). Exchange or displacement of ions already adsorbed by ions of a second electrolyte (cp. Freundlich, ie Verdrangende Ionenadsorption " and Michaelis, "Austausch-Adsorption "). Michaelis (Z. Electrocliem. xiv. p. 353 (1918)) considers that a substance like mastic, or kaolin (bolus), acts as a " zweier electrode " (a binary electrode). Thus kaolin has a slow- moving anion (silicate ion) anchored on its surface and tends to send hydrogen ions into the solution under a definite electrolytic solution tension. Freundlich points out (and Poser, loc. cit.) that other cations can displace the hydrogen ions and form undissociated complexes (and Elissafoff, Z. Phys. Chem. lxxix. p.. 385 1912). (d) An adsorbent which contains some adsorbed electro- lytes need not be necessarily saturated. In this case, besides an exchange of ions, primary adsorption of ions is possible. This also applies to substances which act as binary electrodes in the sense in which the word has been used by Michaelis. He considers that, besides adsorption through exchange of ions, there is only one other type of adsorption, namely, adsorption of both ions in equivalent amounts (" Aquivalent Adsorption"). One other fact has been emphasized by these authors. (e) It is the irreversible nature of electro-chemical ad- sorption. The well known instance of the adsorption of hydrogen sulphide by metal sulphides studied by Linder and Picton (T. Ixvii. p. 163 (1895) ; Whitney and Ober, J. Amer. Chem. Soc. xxiii. p. 842 (1901)) can be mentioned. The adsorbed substance does not come out in solution when the adsorbent is brought in contact with pure water. (/) Lastly, there is no clearly established instance in which hydrolytic splitting up of neutral salts such as potassium chloride has been observed through adsorption. Theories regarding tlie Exchange of Ions. The conception of an adsorbent acting as a binary electrode, suggested by Michaelis, is not of much help in explaining the exchange of ions and other peculiarities of the adsorption of electrolytes. The relationship between the adsorption of ions, electro-endosmotic cataphoresis, and precipitation of colloids has been established beyond doubt. The only theory the Adsorption of Ions. 339 that attempts to correlate them is that due to Freundlich. This view is an extension of Michaelis's idea referred to above. The adsorbent (or colloidal particle) is regarded as a great multivalent ion (dp. Billiter, Z. Phys. Chem. xlv. p. 307 (1003); Duclaux, J. Chim. Phys. v. p. 2\) (1907)). The following extract shows clearly their standpoint (Freundlich and Elissafoff, lor. cit. p. 411) : — "Die Ladung soil nun durch die verschieden grosse Losungs tension der lonen des schwerloslichen festen Stoft's zustande kommen, aus dem das suspendierte Teilchen, bzw. die Wand besteht. Nimmt man als Beispiel das Glas, so hat man an der Oberflache desselben eine Schicht von gelostem, oder bzw. in wasser gequollenem Silkat ; die K- and Na- lonen haben eine orosse Losunostension und bilden eine aussere Schicht, die schwerloslichen, langsam diffundierenden (vielleicht auch stark absorbierbaren) Silikationen bilden eine innere Schicht, die mit dem festen StotF verbunden wie ein vielwertiges Ion sich verhalt. Der wesentliche Unter- schied o-egen ein oewohnliches Ion lieut darin, dass wegen der Grosse GrenzfTachenwirkungen eintreten, die Ronzentra- tion ist in der Umoebuno- nicht so homooen, sondern es sind durch Adsorption hervorgerufene Konzentrationsunterschiede vorhanden. "Fiir zwei ionen gilt nach der Massenwirkungs gesetzt (Anion) . (Ration) = k (nichtdissociertes Salz), deshalb auch fiir das mikronische, vielwertige Anion des als Beispiel betrachteten Glases. " (Vielwertiges Anion) . (Ration) = R (nichtdissocierter Stoff). Es wird also von der Ronzentration der Rationen die ' Ronzentration des vielwertigen Anions/ d.h. auch die Zahl der auf der Grenzflache vorhandenen Ladungen abhangen. "Die Rationenkonzentration, am die es sich hier handelt, wird aber in erster Linie die der nachsten Umoebuno- der Grenzflache, d.h. die Adsorptionschicht sein. Die adsorbierte Menge Ration wird also fiir die ' Ronzentration des viel- wertigen Anions' d.h. fiir die Laduno- der Grensflache niass- . ... gebend sein. Dies ist eine andere Verkniipfung von Adsorption und Potentialdifferenz an der Grenzflache. Genau das Gleiche gilt naturlich fiir ein vielwertioes Ration und die adsorbierten Anionen." There are several difficulties in accepting this theory. Salts of alkali metals can neutralize charged surfaces at moderate concentrations (N/10 or N/20). One has to con- clude that the alkali salts of these " multivalent anions " have a low solubility product. The effect of the valency of the Z2 340 Prof. J. N. Mukherjee on oppositely charged ion cannot be accounted for. The activity of the cations is generally in the following order : — Th > Al > Ba > Sr > Ca > H > Cs > Rb > K > Na > Li. The postulates that alkali metal salts become undissociated at low concentration of the cation and that their solubility products are of the above order for a large number of diverse chemical substances, are contrary to experience. Regarded from the chemical point of view the generality of these ob- servations cannot be explained. Besides, the conception of the suppression of the dissociation of a salt cannot explain the reversal of the charge which is met with when the oppositely charged ion is polyvalent. The view of electrical adsorption put forth by the writer gives a definite correlated account of these various facts. The Role of Electrostatic Forces in the Absorption of Ions. (a) Adsorbent in contact with a single electrolyte : — Let us consider an adsorbent, P, in contact with an electro- lyte A B . It is assumed that the substance P only adsorbs the anion B by chemical affinity. For simplicity it is also assumed that " P " is a pure chemical substance of definite composition. The amount of B adsorbed per unit area will depend on the concentration of A. B and on the strength of the chemical affinities acting on B _ . Corresponding to the number of anions adsorbed an equivalent number of cations A remain in the solution. These are held near the surface by electrostatic forces, and form the second mobile sheet of the double layer (cp. the earlier paper referred to). If the concentration of the electrolyte is sufficient, some of them will be fixed on the surface by electrostatic forces. These ions of opposite charge fixed on the surface by electro- static forces will be spoken of as electrically adsorbed in the sequel. The chemical adsorption of an ion thus concen- trates both ions at the surface in equal amounts. That is, the primary adsorption is an equivalent adsorption of both ions. Analytical methods cannot differentiate between the two adsorptions, but electro-osmotic and cataphoretic experi- ments can (cp. (d) above). If the adsorption of the anion is due to strong chemical forces, perceptible amounts of the electrolyte A B ~ will be adsorbed at very low concentrations. Even saturation may be reached at low concentrations. In such cases, if the the Adsorption of Ions. 341 adsorbent with adsorbed electrolyte is suspended in pure water the adsorbed electrolyte will not be set free (e). Since the primary adsorption of the ions is due to chemical affinity, the influences of the nature of the adsorbent and of the electrolyte (b) are intelligible. (/>) The addition of a second electrolyte : — The general case when both electrolytes, A B~ and C D , are present in all possible concentrations will be too complex. It will be assumed for the sake of simplicity that— (1) the substance P adsorbs chemically the anion B^ strongly, and that the concentration of the electrolyte A B in the liquid is negligible. We are thus dealing with an adsorbent with an amount of adsorbed electrolyte in contact with a second electrolyte solution ; (2) tiie atoms on the surface of the adsorbent P do not exert any chemical affinity on the ions C and D~. This particular case corresponds with most actual systems, and the electrolyte A + B~ plays the part of the " Aktiver Electrolyt " of Michaelis. Let us now consider the effects of the electrostatic forces on the ions C and D~. A cation C , when it- diffuses into the double layer owing to thermal energy, will be attracted to the surface. Considering the kinetic equilibrium between the ions in the second sheet of the double layer (A and C ) and those in the liquid, it is evident that the relative propor- tion of A and ions in the double layer will depend on (i.) their respective concentrations in the bulk of the liquid, and (ii.) their valency. The same consideration applies to the electrically-adsorbed ions A + or C + . At sufficiently large concentrations the whole of the mobile second layer and electrically-adsorbed ions will be formed by the ions C . There will thus be an exchange of ions, and the amount of exchange will depend on the concentration of the second electrolyte. When the displacement is complete the amounts exchanged will be equivalent to the amonnt of B* ions primarily adsorbed and independent of the nature of the replacing ion C + — a fact often observed (cp. iLinder and Picton, loc. cit. ; Whitney and Ober, loc. cit. etc.). The ions will be positively adsorbed. The relationship between the charge of the surface and the positive adsorption of the oppositely-charged ion is also obvious. The amount of () + ions absorbed depends on the 342 Prof. J. N. Mukherjee on amount of the negative ions chemically adsorbed. Anything that increases the total amount of adsorbed negative ions will increase the positive adsorption of C • The reverse case, when the positively-charged ions A + are adsorbed chemically instead of the ions B~? and no other ions are chemically acted on by the surface, is obvious. Negative ions will now be positively adsorbed and exchanged. This state of affairs corresponds with the statements made in (a) and (c) above. Taking the same case again, we shall consider the effect of the electrostatic forces on the ion D • An ion D~ , diffusing into the double laver, will be driven out of it. So long ;is the potential of the double layer is sufficiently strong, a volume of the liquid equal to "SI," where " S " is the extent of the surface and " I " is the thickness of the double layer — will be free from the ion D . In other words, the concen- tration of D~ increases in the bulk of the liquid and a negative adsorption will take - place. This will increase with the concentration of the electrolyte so long as the potential of the double layer is sufficiently strong. Since with in- crease in concentration the potential fall^, the negative adsorption will reach a maximum. At concentrations when the surface becomes electrically neutral, there should be no negative adsorption due to electric forces. It is difficult to determine negative adsorption at high concentrations as the osmotic pressure opposes it. Also, the variations in concen- tration due to negative adsorption become relatively small. The experimental difficulties lie in the analytical estimation of small amounts. Only ions which can be estimated in extremely small amounts are suitable for experiment. Estrup (Koll. Zeitscli. xi. p. 8 (1911)) has actually observed a negative adsorption of the oppositely-charged ion. He estimated the adsorption of the iodate, dichromate, and chromate of ammonium. Michaelis and Lachs (KolL Zeitscli. ix. p. 275 (1911)) did not observe a negative adsorption with potassium chloride. Exactly similar observations have recently been made by Bethe (Wiener Mediz. Wochsch. 1916, Nr. 14 ; Koll. Zeitsch. xxi. p. 47 (1917)). He worked with gelatine gel, gelatine sol, and a number of animal cells. The adsorption of a basic dye is greater in weak alkaline solutions than in neutral solutions. The same is the case for an acidic dye in weak acid solutions. In alkaline solutions the adsorption of acid dyes is negative, and the same is the case with basic dyes in acid solutions. Examples of the role of- the electrical the Adsorption of Ions, 343 force in the adsorption of ions can be multiplied (cp, Baur, Z.Phijs. Chem. xcii. p. 81 (191(5) ; Michaelis and Davidsohn, toe. cit.). Exchange of Bases in Soil and Soil- Acidity It is now easy to understand the nature of the exchange of bases in soil-analysis and the cause of soil-acidity. A com- plete reference to the older literature is given in the following- papers : — (1) McCall, Hildebrandt, and Johnson, J. Phys. Chem. 191b", xx. p. 51. (2) Rxce; ibid. p. 214; (3) Truog, ibid. p. 457. Russell (Brit. Assoc. Rep. 1918, p. 70) has given an excellent summary of the present position of the subject. The facts are that — (a) Neutral solutions of salts like potassium chloride, if treated with samples of soil, give acid extracts though the extract with pure water is neutral. (//) In a large number of cases it has been shown that there is a definite exchange of the cations. Equivalent amounts of bases are exchanged in many cases. Two different views have been advanced to explain the facts, The older chemical view regards the process as a chemical interaction between definite acids (e. g., humus acid) or complex salts {e.g., silicates) and salt solutions. The other view begins with von Bemmelen, and regards it as an adsorption process. Cameron suggested (cp. Russell's Report) that the soil adsorbs the base more strongly than it adsorbs the acid. The objections against the chemical view can be sum- marized as follows : — The extract wich pure water being neutral, the soil-acids must be insoluble. The acids must be unusually strong, as they evidently decompose a neutral salt solution combining with the base, liberating the strongest known acids, like hydrochloric acid. Evidently such acids are unknown, and it is difficult to conceive of such reactions. Regarding the exchange of buses, the difficulty lies in the assumption that the basic ion is taken up to form an insoluble salt. It is necessary to postulate the existence of insoluble salts of alkali metals in a large number ot' cases (cp. the remarks on Freundlich's theory). That adsorption plays an important part is also evident from the works of: Russell and Prescott (J. Agric. Sci. viii. p. 05 (191G)) on the interaction of dilute acids and phos- phates present in the soil. But the view of: Cameron does 344 Prof, J. N. Mukherjee on not seem to be tenable. The preferential adsorption of an ion by the soil does not mean hydrolytic decomposition of the salt. It appears from the summary given by Russell •that the equivalent exchange of bases lies in the way of regarding the reaction as an adsorption process (loc. cit. pp. 71, 75 ? 76). It would be apparent from the previous discussion that this, in itself, does not contradict the adsorp- tion hypothesis. Soil can be regarded as a complex colloidal system. It is a complex gel consisting of aluminium and other silicates, free silica, ferric hydroxide, etc. The gel is mixed with insoluble crystalloids. It also contains small quantities of adsorbed electrolytes and organic matter in indefinite and varying proportions. The gel adsorbs anions by chemical affinity. These anions may be : — (1) of organic acids, such as humus acid ; (2) of simple electrolytes like chlorides, sulphates, car- bonates, etc. ; (3) hydroxyl ions from water. . Owing to the complex chemical nature of the gel and the enormous specific surface of gels, large quantities of anions may be adsorbed. An equivalent number of cations remain near the surface as the mobile second sheet or as electricalty adsorbed. The exchange of bases is simply due to the dis- placement of these ions. When the displacement is quantitative equivalent amounts are exchanged. The anions primarily adsorbed or the cations in the second sheet are not of one kind. The relative numbers and chemical natures of these ions will evidently vary with the different soils. An extract with pure water will be neutral unless the soil contains free acids. An extract with a neutral salt can only be acid when the cations displaced from the second sheet (or electrically adsorbed) contain hydrogen ions or such ions as aluminium, which hydrolyse in dilute aqueous solutions. The role of the aluminium ions in determining the acidity of the soil extract has been pointed out by Daikuhara (Bull. Imp. Central Agri. Expt. Station, Tokio, ii. pp. 1-40 (1914)), and has been fully confirmed by Rice (loc. cit.). The function of organic acids lias constituted a great objection against the adsorption hypothesis. The hydrogen ions in the second sheet have probably, in most cases, their origin in these acids. This view thus correlates the exchange of bases observed with soil with such exchanges as have been observed in the adsorption of electrolytes (cp. Michaelis). That sometimes considerable quantities of bases are ex- changed should be referred to the enormous surface of these the Adsorption of Ions. 345 gels, and that probably the surface is saturated with anions. As crystalloids (insoluble) are also present, the type of exchange considered by Paneth (loc. cit.) is also possible. It is needless to point out that in this discussion only the theoretically simple case has been considered. Complications due to simultaneous primary adsorption of different ions and their mutual displacement are not always negligible. Besides, the changes may not be restricted to the surface ; formation of solid solutions, etc., are not excluded. Considering all these complex influences, it is interesting to note that most of the observed regularities correspond to the theoretically simple case. Adsorption of Ions in its Relation to Permeability of Membranes and to Negative Osmosis. In conclusion, a few remarks will be made on the funda- mental interest that a study of the adsorption of ions has for biological phenomena. Cell activity is greatly conditioned by the permeability of its "walls" or the cell-substance to the contents of the liquid with which it is in contact. The connexion between the rate of osmotic flow through mem- branes and even the direction of the flow, and the potential differences existing on the two sides of the membranes, has been clearly established (Girard, C.R. cxlvi. p. 927 (1908), and following authors : Bartell, J. Amer. Chem. Soc. xxxvi. p. 6±6 (19.14) ; Hamburger, Z. Pliys. Chem. xcii. p. 385 (1917)). The origin of the potential difference is generally assumed to be due to the fact that the rate of diffusion of the electrolytic ions in the membrane substances is different from that in water. That the membrane potential is due to a selective permeability of ions was first suggested by Ostwald (Z. Phys. Chem. vi. p. 71 (1890)) ; Donnan (Z. Elektrochem. xvii. p. 572 (1911)) has discussed the origin of the potential differences theoretically, and has given it a quantitative form based on thermodynamic considerations. In collaboration with others he has carried out a number of investigations which have established the validity of this view. The simpler case of a potential difference between two interfaces when an immiscible liquid is placed between two aqueous solutions has also attracted a good deal of attention. The work of Loeb and bis co-workers on cell- permeability and origin of the membrane potential is of fundamental importance (Loch and Beutner, Biochetn. Zeit. li. p. 295 (1913) ; Beutner, Z. Phys. Chem. lxxxvii. p. 385 346 Prof. W. M. Hicks on certain Assumptions in the (1914), Z. Elektrochem. xix. pp. 329, 473 (1913) ; Loeb, J. Gen. Phys. xx. p. 173 (1919), ii. pp. 273, 255, 387, 577, 563, 673, 659). The part played by the adsorption of ions in these phenomena is twofold. The origin of the potential is in many instances due to the adsorption of ions (cp. Baur, Z. Elektrochem. xix. p. 590 (1913) ; Z. Phys. Chem. xcii. 1916, p. 81). Secondly, the electrostatic forces of the surface probably determine the relative permeabilities of the two ions. To this the semi- permeability of an ion can be referred. Regarding negative osmosis, attention may be drawn to the suggestion of Freundlich [Koll. Zeitsch. xviii. p. 1 (1916)) that the thin walls of the membrane substance conduct electricity, and electro-osmotic flow of the liquid occurs. A necessary condition is that one ion is permeable and the other relatively impermeable. This explanation meets thermoclynamical requirements, and is the only satis- factory one hitherto put forward. In all these cases the same influences of polyvalent ions and ions of opposite charge are noticeable. The change in the collodial properties of the membrane is an important additional factor which has to be remembered. The influence of the electrostatic forces is unmistakable. Physical Chemistry Department, University College, London. XXX. On certain Assumptions in the Quantum-Orbit Theory ' of Spectra. By W. M. Hicks, F.R.S* THE practically complete success of the quantum-orbit theory in describing all the known facts of spectra, in cases where we know experimentally that the source consists of a single nucleus and a single electron, must give assurance that the same procedure must also be capable of application to more complicated atoms than those of the hydrogen and enhanced helium types. Unfortunately, however, mathematical difficulties have so far prevented any rigorous application of the theory to definite cases, even of the next simplest atomic configuration of a single nucleus and two electrons. The attempt of Sommerfeld at an approximate solution shows, on the one hand, how hopeful we may be of a description of spectra on this basis, and at the same time how far we are at present from its * Communicated by the Author. Quantum^ Orbit Tlieory of Spectra. 347 achievement. In the present note I wish to illustrate this by drawing attention to certain assumptions as to actual spectral data, which have been made and which do not appear to be justified. The criticisms may not affect essential points, but they would appear to require some modification in the presentment of the tlieory. References will be made to Sommerfeld' , s 'Atombau und Spektrallinien,' 2nd edition (1921). 1. Sommerfeld (pp. 27G, 50b') takes a configuration of a central nucleus, surrounded by a ring of equally-spaced electrons, and at a considerable distance furl her out one electron revolving in a quantized orbit. On fhe assumption — here justified — that the ring can be treated as if the w r hole charge of the electrons on it were continuously distributed along it, he obtains as an approximation the same form for a sequence function (or term) p as that' suggested by Ritz, viz.* : He says that this is the actual true form, as already deter- mined by observation. This is, however, by no means the case. No form has yet been found which will fit in for all series, and indeed the form N/(m + /u. + a/m) 2 is in general rather superior to that of Ritz. It is to be noted that the assumption made above leads to the same result as if the force to the centre depended only on forces inversely as even powers of the distance, and forces depending on odd powers — say l/r' d — are excluded. It may also be noted in passing that the theory so developed applies only to the case of a single external electron and one internal ring, that is, on the usually assumed configuration of eight- electron rings, only to the spectra of the fluorine group, or the ionized rare gases, or the doubly-ionized alkalies, etc. By taking his E as (k—sj^e in place of ke, the formula would meet the more general case. This modification, however, would only slightly affect the order of magnitude of the quantities /a, a. In the formula m=n-\-n', where n, n are respectively azimuthal and radial quantum numbers, and //,, a arc functions of n and not of n'. * As a result of successive approximation, « being small, -this means for a complete approximation the form which, as is well known, is capable of reproducing- practically all cases if /*, a, {3 . . . are all at disposal, and are not related necessarily to one another, as here. 348 Prof. W. M. Hicks on certain Assumptions in the It is not to be expected that the numerical values of the constants fju, a. on this special theory should accord with any determined by experiment, but they should be of a suitable order of magnitude and general character. It may be interesting to test this. The expressions for the constants- //,, a may be written 8 ( ,15/3/ 3*-*A ,1 2 where , _ (27r) 4 m 4/< and <x 3?r ^ = (2,)%Ve ( z^ ) = 8 . 9(Z _^ )( ^^ )ioiv /9_Z-*x 2 /* " 2n 2 h\2 k-sic) 7 ' If p be measured in wave number instead of frequency, the a must be multiplied by the velocity of light. Then uc 6-12 /9 Z-£ /9 &-k \ \2 k-sj /jl n" \4 /c — Sk Here r denotes the radius of the internal ring in cm., Z the atomic number of the element, k the number of external electrons, and s k depends on the mutual action of the external electrons on one of them. For Li, Z = ll, & = 3, s/ >; = , 577, and 2-2/. 4-5 10 16, In actual cases, for wave numbers of about p = 10 5 , poLc/ji lies between about *9 and *01. Hence the second equation requires r< 10~ 7 ' 5 > 10 -8 ' 5 . Since fi<l, the first equation requires r to be about 10 -8 , but as the second term in the bracket is determined by an approximation it must be a small fraction, whence r < 10 -8 ' 5 . The fact that both give values of the same order of magnitude, even if they cannot exactly agree, and not far from what might be expected for an 8-electron ring, is certainly satisfactory. 2. It is deduced that the different types of sequences correspond to azimuthal numbers n — 1, 2, 3, 4, the different orders of the same type to radial quanta n' = 0, 1, These are then co-ordinated with the s, p, d, f types because it is stated that these types have their lowest orders respectively of 1, 2, 3, 4. It is difficult to see how this statement has been arrived at, as it is quite incorrect. For the sake of Quantum- Orbit Theory of Spectra. 34',) readers who may not be familiar with spectral data, it may be well to consider them here. (s, p.) For the s, p the lowest order are : s. p. TT [ Alkaline earths ... 2 1 "'iZn, Cd, Eu, Hg... 2 1 p. Rare gases 1 1 T [The alkalies I 2 L - \Cu, Ag, Au 1 J Group III 2 1 I£ it were not for the cases of the rare gases and the Cu subgroup, the assumption might be explained by an interchange of the nature of the sequences which produce P, S series (for which in Groups I. and IJ. indeed there is also direct evidence). But that two groups make s, p both have unity for their first order is fatal. (r/.) The assumption of 3 as the first order for d(in) no doubt is based on the fact that Pitz made it in dealing with the D series in the alkalies. The denominators of the first orders in this group are comparable with 2*9, which Ritz wrote as 3 — '1 and called the first order 3. But this procedure is inadmissible either on the side of the formula or from what Ave know of the constitution of the d sequence. In Sommerfeld's formula /jl is positive, and it is only by treating the fraction as positive that we find a detinite dependence of it on certain spectral constants. But even so, the first order is not 2 for all groups. The law of the first order of the d sequence is a quite simple and definite one, and is given on p. 18S of my recently published 'Analysis of Spectra.' It is that in each group of the periodic series, the subgroup of elements whose melting-points increase with atomic weight take as their first order m = l, whilst the sub- group with decreasing melting-points take m = 2. (/.) In the case of / the 3 + fraction has again clearly been written 4 — /, and the assumption lias been made that the lowest orders of the / type take m = 4. But here also, for the same reasons as in d, the fraction must be taken as positive. In the alkalies certainly the lowest order observed is F(3), but F(2) would lie far up in the ultra-red, beyond even Paschen's longest lines. In the Cu subgroup there is evidence for m = 2 and indications for m = l. The alkaline earths have m — 2 both in triplets and doublets. In the Zn subgroup only F(3) has been observed, but F('2) would lie in the extreme ultra-red. In Group III. there is no evidence, whilst in the rare gases there are examples of* F(l) and F(2). It would thus appear that the theoretical deduction that different types depend on successive changes of azimuthal quanta by unity is not tenable. 350 Prof. W. M. Hicks on certain Assumptions in the 3. In dealing with the Zeeman effect on p. 422, Sornmer- £eld adopts Paschen and Back's interpretation of their experiments on the Zeeman effect in the case o£ close multiple lines. This interpretation was based on precon- ceptions as to the nature of the series types in He and Li, which they investigated. I have given * reasons why this interpretation should be modified. On either inter- pretation, however, a consequence follows which appears difficult to explain on the quantum-orbit theory. Take, for example, the case of the helium doublet at 4713 A. Each component in weak fields shows special Zeeman patterns. W4th increasing fields and consequent approximation of certain constituents from each pattern, an interaction occurs of one on the other. Such an effect can only be produced if the two patterns are produced in the same source. Hence the original components of such a doublet must be produced simultaneously in each atomic configuration, whether a magnetic field is present or not. It follows that in radiation there must be simultaneous passages of two electrons, each from its original orbit to its final one. But as the effect takes place at one operation, the total change of energy is passed on to the radiator and emitted as a single mono- chromatic radiation, i. e. no doublet. It might be suggested that the effect could be explained on the hypothesis that the magnetic field affects the mutual possible orbits, and that sometimes one passage occurs and sometimes the other. It is difficult to see, however, how an orbit can be modified by another supposed one which is non-existent, i. e. not being- described at the same time, 4. This consideration does not affect evidence for the quantum theory, but will serve to illustrate a habit which is somewhat exasperating in reading the writings of many exponents of the quantum theory — viz., the picking up of small and often irrelevant points as charming results of the theory. On p. 300 ff. it is expected that each doublet separation on passage from arc to spark conditions should be magnified in a measure corresponding to the ratio 4N:N, and satisfaction is expressed that in data adduced from corresponding elements in the doublets of group I. and the enhanced doublets of group II. this expectation is fulfilled. The ratios of the separations are reproduced (with Hg : Au added) in the first line of the following : — Mg. Ca." Sr. Ba. - 5-3 3-9 3-4 3-1 2-24 210 2-03 1-92 * 'Analysis of Spc^tr.i/ § 7, p. 96. Zn. 3-5 ca. 2-7 Eu. 9 2-5 2-21 242 2-03 0.]9 Quantum^ Orbit Theory of Spectra. 351 But surely these numbers show that the comparison is not justified. As is known, a correspondence actually lies between the enhanced doublets and the triplets in the same element. Thus the ratios of the doublets to the first separation of the triplets are given in the second line of figures above, where the agreement is remarkably close. In this latter case, however, the correlation is not direct. It is due to three concurrent facts : (1) ratio 4N : N ; (2) the oun multiples in the doublets and v\ of the triplets are very nearly the same in each element ; and (3) the denominators in the doublets and triplets have nearly the same ratio in all (see belowj. There is, however, a close correspondence between the mantissse of the doublets in the two groups I. and II., especially as between the alkalies and the alkaline earths, those of the latter being about double the first. Correlation is also shown between" the denominators of the triplet and doublet sets in all the group II. elements. These statements are illustrated by the following data : — II. I I. Tripl. Doubl. Ratio. Doubl. Ratio Mg ... ... 1-660 2-265 1-36 Na .... . 2-117 2-27 Ca ... ... 1-796 2498 1-39 K . 2-235 2-12 Sr ... ... 1-880 2611 1-39 Kb .... . 2-292 2-09 Ba ... ... 1-957 2735 1-39 Cs .... . 2-361 2-03 Zn- ... ... 1-599 2-098 1-31 • Cu .... . 1-869 Cd ... ... 1641 2-144 1-30 Ag -. . 1-892 Eu ... .. 1-648 2190 1-32 Hs... .. 1-653 ? Au .... . 1-929 Here 'under II. the third column gives the ratio of the denominators ; under 1. the second column gives tlie ratio oj mantissa' in JLl. to those in I. On the other hand, there appears very little correlation between the oun multiples which give ,the separations in corresponding elements of groups I. and II. A much closer one is found between those of the triplets and doublets of the same element in II. Thus, in the following, the first line gives the ratios of the oun multiples of doublets in group I. to those of the enhanced doublet of the corre- sponding element in group II. The second line gives the ratios of these multiples for the first triplet separation and the enhanced doublet in each element. Mg. Ca. Sr. Ba. Za. Cd. •683 •779 •843 •903 •847 11 •391 •709 •726 •761 •789 •846 Eu. •842 He-. [ 352 ] XXXI. On the Theory of the Characteristic Curve of a Photo- graphic Emulsion. (Communication No. 22 from the British Photographic Research Association Laboratory.) By F. ( !. Toy, M.Sc, F.Inst.P., F.R.P.S. * IN the most recent investigations on the relation between the photographic effect and the light-exposure, special plates containing only a single layer of grains have usually been employed. With such plates the photographic effect is determined by counting the percentage of grains made developable. The curve expressing the relation between this percentage (x) and the logarithm of the exposure may be called the characteristic curve of a single-layer emulsion, corresponding to the ordinary curve of a commercial emulsion, in which, instead of x, values of the density (in the photographic sense) are plotted. In a recent paper (Phot. Jour. 1921, lxi. p. 417) the author has shown that such a curve, for a set of grains which are geometrically identical, is of the usual S-shaped tj^pe, i. e. a difference in size or shape does not account for the fact that all the grains do not become developable with the same ex- posure. Now, a set of geometrically identical grains, all in a single layer and similarly orientated to the incident light, represents the simplest possible emulsion which we can in- vestigate experimentally. It also corresponds to the simplest theoretical case, eliminating many complicating factors which, though greatly affecting the form of the characteristic curve, have nothing to do with the primary mechanism of the photograph process. In other words, with this emulsion the curve is reduced to its " purest " form, and is determined almost solely by the photochemical process which takes place. It is now generally believed that the primary action of light on the grains is to form in or on the surfaces of them certain " centres " or " points of infection " which act as starting-points for their reduction by the developer. This view has for some time had considerable evidence in its favour. Chapman Jones (Phot. Jour. 1911, li. p. 159) showed that by stopping development at a very early stage it is possible to get particles of silver too small to be visible microscopically, but which can be shown to be present by the colour imparted to the film, and by enlargement to visible dimensions by the deposition on them of mercury. Hodgson (Brit. Jour. Phot. 1917, p. 532) carried development a little * Communicated by Prof. A. W. Porter, F.R.S. Characteristic Curve of a Photographic Emulsion. 353 further, and showed it possible to observe the silver reduced by the developer only around certain centres in the grain, and a recent paper of Svedberg's (Phot. Jour. April 1922) leaves little room for doubt that the possibility of a grain being made developable depends on the existence in it of some kind of reduction centre. Opinion as to the nature of these centres seems at present to be divided. There are those who assert that they are formed by the light-action, and that they do not exist before exposure is made. Such, for example, is the case if the centre is really a molecule of silver halide which has lost an electron, as is believed by H. S. Allen (Phot. Journ. 1914, liv. p. 175). On the other hand, there are those who believe that the centres are actual particles other than silver halide formed in the grains during precipitation and subsequent ripening, and that these only become susceptible to the action of developer after exposure to light. There certainly is considerable evidence to show that silver halide is not the only substance in the grains. Luppo- Oramer (Kolloidchemie una 1 Photographie) was led, as a result of his work, to the conclusion that, at any rate in the most sensitive emulsions, nuclei are present which probably consist of a colloidal solution of silver in the halide. Renwick (J. S. C. I. 1920, xxxix. No. 12, 156 T.) extends this idea, and says : " In our most highly sensitive photographic plates we are dealing with crystalline silver bromide in which, besides gelatin, some highly unstable form of colloidal silver exists in solid solution, and it is this dissolved silver which first undergoes change on exposure to light." These silver particles are negatively charged, and Renwick believes that the action of light is to discharge, and hence to coagulate into larger groups, those particles of colloidal silver which existed in the grain before exposure ; it is these groups of coagulated electrically neutral particles which are the re- duction centres. This view is supported by the ultra-micro- scopic observations of Gralecki {Koll. Zeit. 1912, x. pp. 149- 150), who showed that X-rays have a coagulating effect on the particles in gold sols ; by Svedberg {Koll. Zeit. 1909, iv. p. 238), who has similarly shown that ultra-violet light agglomerates ultra-microns to larger aggregates; by Spear, Jones, Neave, and Shlager (J. Ainer. Chem. Soc. 1921, xliii. p. 1385), who have observed the same kind of effect with colloidal platinum; and by recent experiments of Weiger and Scholler {Sitz. Preuss. Akad. Wiss. Berlin, 1921, pp. 641-650). PA/7. Maq. S. 6. Vol. 44. No. 260. Aug. 1922. 2 A 354 Mr. F. C. Toy on the Theory of the These facts are at any rate sufficient to justify an attempt to explain the relation of the number of grains changed to the light-intensity on the basis of the existence in the grains of: actual particles which are not silver halide. These are not necessarily all changed to reduction centres from in- active particles by the same light-energy as they would be if they were single molecules of the halide. We shall make no assumptions as to the composition of these centres, and the theory does not depend on their being composed of colloidal silver. We shall use the term " nucleus " rather than centre to indicate the presence in the grains of actual particles before exposure. Ihe Characteristic Curve of a Set of Geometrically Identical Grains. llieoretical. Our first object is to consider the case of a set of grains of identical size and shape, and to determine the relation we should expect to find between the percentage of these which are made developable and the light-intensity. The time of exposure is kept constant throughout. If we consider a volume V of the silver halide which is very large compared with that of a single grain, we may assume that the total number of nuclei in any such volume of the emulsion is the same, though the number contained in individual grains in this volume may vary. We will define the sensitivity of a single nucleus as the minimum intensity which must be incident upon it in order to make it " active " in the presence of the developer. For a given intensity of the incident light there will be a definite number of such active nuclei in every volume V, and they will be distributed amongst the grains entirely haphazard, according to the laws of chance. Every grain which happens to have at least one active nucleus will be developable. When the intensity of the light increases, more grains are changed. On any " nucleus " theory this happens because. more nuclei are present, so that a single grain has a greater chance of having at least one of them. This may be ex- plained in one of two ways. Firstly, all nuclei may have the same sensitivity, say I, but owing to the rapid absorption of light, those nuclei which are situated in the grain at some distance from the surface on which the light is incident, do not receive an intensity of I when the incident intensity is small. As the latter becomes greater, the volume of silver halide, throughout which the intensity is at least I, increases, Characteristic Curve of a Photographic Emulsion. 355 so that the number of active nuclei increases also. Secondly, the sensitivity of every nucleus may not be the same, so that as the intensity of the light is increased, nuclei become operative which are unaffected by lower intensities, and again the total number of active nuclei increases with the intensity. We will consider only the case of grains in the form of thin plates as they occur in high-speed emulsions. Eggert and Noddack (Preuss. Akad. Wiss. Berlin. Ber. 1921, xxxix. p. 631) have recently measured photometrically the fraction of the incident light which is absorbed by an ordinary commercial photographic plate, and have found it to vary with the different plates from about 4 to 12 per cent, for violet light, for which the amount of light absorbed is near the maximum. Now, these plates contain several layers of grains, so that a very extreme upper limit to the fraction of light absorbed by a single grain is, say, 20 per cent. Thus, if there is an increase in the incident intensity of the order of 20 per cent., the intensity of the light transmitted through a grain will be equal to the intensity incident before the increase took place. Thus, if all nuclei are equally sensitive, a change in the incident intensity of the order of 20 per cent, will cause a difference in the number of active nuclei from zero to some fixed maximum, so that the characteristic curve can only function over a range of intensity such that the ratio of its extremes is of the order of 1*2 : 1. As will be shown later, for the steepest characteristic curve plotted this ratio is about 25 times as much as this, so that as an appreciable factor in determining the increase of nuclei with intensity the first assumption is untenable. We have, therefore, to assume that all the nuclei are not equally sensitive. Since these nuclei are all formed in the same emulsion, most of them will have a sensitivity near the average value for the whole, and there will be a few which are very sensitive and a few which are very insensitive. There will be none which will respond to zero intensity, and none so in- sensitive that it takes an infinite intensity to affect them. We therefore expect the curve showing the relative number of nuclei R having any given sensitivity I to be of the general form shown in fig. 1. The exact mathematical form of this curve is immaterial at present, but it will be similar in general form to that obtained by Clerk Maxwell for the distribution of velocities between the molecules of a gas. By similar reasoning to his, the number of nuclei (Nj) which 2 A2 356 Mr. F. C. Toy on the Theory fo the have sensitivities between zero and I x (which is the number operative when the intensity of the light is Ij) is given by Fig. 1. o i, the area OAB, i. e. i-» N,: J/ a)dI ' (1) where /(I) gives the values of the ordinates in terms of I ; or if N is the average number of nuclei per grain and a the number of grains in volume V, N, ■-;J>.*. . . , (2) The total number of nuclei is given by N=f/(I).c*I. Fig. 2. The curve showing the relation between N and I, shown in fig. 2, is characterized by its unsymmetrical-shaped S form. Characteristic Curve of a Photo<jrapliic Emulsion. 357 When the a grains in the volume V of silver halide are subjected to an intensity I t , every grain which happens to have at least one of these N : nuclei will be made developable. We have, therefore, to find the chance of a grain containing at least one of the N\ nuclei when they are distributed hap- hazard amongst a grains. This can easily be obtained from the theory of probability. If p denotes the very small probability that an event will happen on a single trial, the probability P r that it will happen r times in a very great number, say n trials, is (Mellor, 1 Higher Mathematics/ p. 502) V r =(n P y.e-»P/rl (3) Let the volume of a single grain be v, then since the volume of every grain is the same the total volume V is an. Let p be the very small probability that a volume dv will contain a nucleus, then p = ^ 1 .dvlav (4) To obtain the probability of the volume v containing a nucleus, we may suppose each dv to be a trial, so that the number of trials n is n = vjdv (5) Therefore the value of np in equation (3) is Nj/a, which is equal to N . If in this number of trials the event (i. e. v containing a nucleus) happens once, a grain will contain one nucleus; if it happens r times it will contain r nuclei, so that from (3), (4), and (5) we see that the probability of a grain containing r nuclei is p r =(N y*-^/H, (6) which is the same equation as was obtained independently and first published by Svedberg. The probability of a grain containing no nuclei is the value of this expression when r = ; i. e. Now, since it is certain that a grain must contain either zero or at least one nucleus, the probability P x that a grain will have at least one is P 1 = 1_,-n (7) 358 Mr. F. C. Toy on the Theory of the But if x\a is the fraction of grains which are changed, F 1 = xja ; or, denoting log a/ (a — x) by A, we have A = N . Thus the same form of curve should be obtained when N is plotted against the intensity as is obtained when A is plotted against the same variable. The form of this curve should be an unsymmetrical S, as shown in fig. 2. Experimental. The first experiment carried out was to determine the relation between A and the light-intensity I for a set of geometrically identical grains, every grain counted being measured, as described in a previous paper (Phot. Jour. 192 L, lxi. p. 417). Table I. Cross-section of grain = 0*98/x, 2 . Log I (relative). X. X (curve- values.) A. Log I. X. X (curve values.) A. 92-0 91-5 2-45 -0-893 47-7 466 0-62 -0162 89-5 90-0 2-30 -1-215 5-2 5-2 0-05 -0-310 87-7 87-0 2-04 -1-487 o-o o-o o-oo -0-572 74-8 750 1-39 -1-788 00 o-o 0-00 In the first two columns of Table I. are shown the values of log I and x determined experimentally by exposure behind a neutral wedge, and these are plotted in fig. 3. The values of x given in column 3 are read off the curve in fig. 3, and it is these values which are used in calculating A in column 4. This is the best way of obtaining the A values, since when x is large a very small error in its determination means a very large error in A. The A, I curve, shown by the solid line in fig. 4, is exactly as predicted by the theory. We must note here that this is not in agreement with the results of Slade and Higson (Proc. Roy. Soc. 1921, A, xcviii. p. 154) and a previous experiment of the author (ibid. 1921, c. p. 109) to confirm their result. Slade and Higson stated that the relation between A and I can be expressed by the equation Arrairi-^ 1 ), Characteristic Curve of a Photographic Emulsion, 359 where a and ft are constants. A comparison of the form of this curve (fig. 5) and the curve in fig. 4 shows a difference Fiff. 3. i A ► T O / -" ? / / / / / / 1 I / / 0-2 04 0-fc 0-8 1-0 360 Mr. F. C. Toy on the Theory oj the when I is large, but this can be explained. Firstly, the grains used in Slade and Higson's experiments were not all of one size, the variation being about 30 times that in the present case. Secondly, the best curve given in Slade and Higson's paper has actually the same form as that in fig. 4 if equal weight is given to each point plotted. Also, in the author's confirmatory experiment, the main point was to show that at low intensities A varied, at any rate approximately, as I 2 ; at high intensities the work was not nearly as accurate Fig. 5. I— > as in the present case. To be certain of the form of the curve in fig. 4 the upper part of it was plotted for another size of grain, and that the same result was obtained is shown by the dotted line. The position of a nucleus can be detected by Hodgson's method of partial development of the exposed grains (ibid.). The developer used was made up as follows : — 200 c.c. saturated Na 2 S0 3 , 8 c.c. 10 per cent. KBr, 0*3 gm. Amidol. This is a weak, slow developer, and is best for this purpose because there is a bigger latitude than if the developer is strong in the time of development necessary to render the position of the nuclei visible and yet distinct from one another. The best development time was found by trial and examination of the grains under the microscope. After exposure the plate was plunged into the developer for a known time, then quickly and thoroughly washed, and dried without fixing. The flat triangular grains used were so thin that the silver deposit was visible without dissolving away Characteristic Curve of a Photographic Emulsion. 361 the silver bromide. In fig. G are given some examples of grains in which these nuclei appear ; they are formed more on the edges of the grains than anywhere else, though quite a number appear either inside or on the flat surfaces. Fig-. 6. Positions of nuclei numbered. The next experiment was to show if equation (6) holds good. A plate was exposed to a uniform intensity, partially developed, and the number of nuclei occuring on each of 150 grains was counted. Hence the average number per grain was known and also the number of grains having 0, 1, 2, 3, etc. nuclei each. In Table II. are given the theoretical and observed values of P,. for two equal-sized sets of grains, in one and the same emulsion, having widely different values of N . Table II. N = 0-180. N = 1-193. Value of r. No. grains having r nuclei. Probs Obs. ability Pr. Calc. Value i of r. No. grains having r nuclei. Probability Obs. Calc. 91 O607 0619 i 43 0-287 0303 I 47 0313 0297 1 55 0-367 0362 2 11 0073 0071 2 36 0240 0-216 3 1 0007 0001 3 12 0-080 0-086 4 0000 0001 4 4 0-027 0-026 The observed values of P r were determined by the fact that the probability of a grain having r nuclei is equal to the fraction obtained by dividing the number of grains which have r nuclei by the total number, i.e. 150. In fig. 7 the theoretical values are represented by the smooth curves, and 362 Mr. F. C. Toy on the Theory of the those observed by the plotted points. The agreement is very good, and proves the validity o£ equation (6) in the case of a fast emulsion. Ker. 7. 0-8 Ob r ro-4 0-2 1 1 >< 10 N = 0-480. W N Q =1-H3. \ \ \ ^ K u \ - \ \ X ^) > i r __J- 5 To find the relation between the average number of nuclei per grain and the intensity, a plate was exposed behind a step wedge and partially developed. The size of grain selected was the same as used for determining the A, I curve in fig. 4, the plate being exposed for approximately the same time behind the same wedge. At each intensity (I) the total number of nuclei on 200 grains was counted (except at 1 = 0*044, where 100 grains were considered sufficient), and hence the average number per grain found. The values are given in Table III., and it will be seen that the curve in fig. 8 is of the same general form as the A, I curve in fig. 4, as is predicted by the theory. Table III. Cross-section of grain = 0'98/u, 2 . I. N . L N . 1-000 0-98 0-270 0-24 0-689 0-88 0-180 005 0-490 0-63 0-128 0-02 0-356 43 0-044 o-oo Characteristic Curve of a Photographic Emulsion, 303 The highest value of N is about 1, which corresponds to less than 70 per cent, of grains changed, whereas actually the percentage changed corresponding to this value of N was about 90. Tbis is because the partial development has not been sufficient to show up all the nuclei, and it is very difficult to do this, since before this stage is reached, nuclei which initially were distinguishable from one another have grown together into a single mass of silver. It is, however, very unlikely that even if every nucleus could be observed the general form of the curve in fig. 8 would be changed. Fte. 8. +00 N, O QSfr op 025- -e- 0-2S 0-50 0-75 ■00 The most natural assumption to make is that longer develop- ment would merely result in an increase of the number of visible nuclei in proportion to the number already observable, and that this is the case is shown by the following experi- ment : — Two plates were given the same exposure under the wedge and partially developed, one for 15, and the other for 18 seconds. The values of N were then found for widely different intensities, with the following results : — (1) I = 1*00, N for 15 seconds development = 0*613, N for 18 seconds = 0'980, whence (N ) 18/(N )15= 1-59. (2) I = 0*27, (N )15 = 0-153, (N )18 = 0*240, whence (N ) 18/(N ) 15 = 1*57 ; so that this ratio is practically constant, and the general form of the curve is indepen- dent of the development. 364 Mr. F. C. Toy on the Theory of the Variation of Grain Size. Experimental. It will be convenient to deal first with the experimental curves. When the values of x were being found for the curve in fig. 4, the corresponding values for three larger sizes of grain were determined at the same time and in the same way. The characteristic carves for the four sizes are shown in fig. 9. Fig. 9. Logl-> The important points in regard to these curves are that for one and the same emulsion : — (1) a set of large grains is more sensitive than a set of small ones, which confirms Svedberg and Anderson's result (Phot. Jour. 1921, Ixi. p. 325) ; (2) the characteristic curve for small grains has a greater maximum slope than that for large ones, i. e. } the ratio of the intensity which just changes all the grains to that which just causes the smallest possible change is larger the larger the grain size. As will be seen from the figure, the logarithm of this ratio for the smallest size grain is about 1'5, which is equal to an intensity ratio of 30 : 1, whilst for the largest size a ratio of 100 : 1 is necessary to give half the curve. In Table IV. are given values of x, as read off the experi- Characteristic Curve of a Photographic Emulsion. 365 mental curves in fig. 9, corresponding to known relative intensities, and in the third column the values of: A are Table IV. Cross-section of (a) =0'98yu, 2 , (6) = 1-75 M 2 , (c)=2-73/, (<0-8-9V- X (curve values) . A. Relative Intensity. 1-000 (a). (ft). («). (d). (a). (b). (')■ (<*)■ 91-5 95-8 97-5 98-0 2-45 3-16 3-70 391 0-689 900 94-5 96-8 97-8 2-30 2-91 3-45 3-80 0-490 87-0 93-0 96-0 97-0 2-04 2-66 3-22 3-51 0270 75-0 87-6 92-5 94-5 1-39 2-08 2-60 2-90 0-128 460 73-0 81-0 88-5 0-62 1-31 1-66 2-16 0-06 I 5-2 52-5 66-6 78-0 0-05 0-74 1-10 1-51 0-033 00 36-0 54-0 67-6 o-oo 0-45 0-78 1-12 0016 00 20-7 40-5 570 o-oo 022 0-52 0-84 0-008 o-o 70 28-5 46-5 o-oo 0-07 0-33 0-62 Fig. 10. T A* s / // / / ^^T ^^^r t 0-25 0-50 075 X- 1-0 calculated. The A, I curves for the four sizes of grains are given in fig. 10. © © 366 Mr. F. C. Toy on the Theory of the Iheoretical. Consider what is the effect of a variation in grain size on the nuclei distribution curve shown in fig. 1. We will first assume that the sensitivity of a nucleus is quite independent of the size of the grain in which it chances to be, i. e. once a nucleus is formed in a grain, its sensitivity does not change as the grain grows. This is apparently Svedberg's assumption, for he says : " the small and the larger grains in one and the same emulsion are built up of the same kind of light-sensitive material — -just as if they were frag- ments of different size from one homogeneous silver bromide crystal/' If this is the case, then the only result of in- creasing the size of grain is to increase the total number of nuclei, and these will be distributed amongst the different sensitivities in the same proportion as before. This is shown in fig. 11, where the distribution curves for two sizes of Kff. 11. grain are given. We have made no assumption regarding the relation between total number of nuclei and grain size except that large grains have more than small ones. The curves relating I and N (average number of nuclei per grain) which will be obtained from distribution curves such as those in fig. 11 are shown in fig. 12. We have already shown that the N , 1 curve is identical in form with the A, I curve, so that those in figs. 12 and 10 should be of the same form. As a matter of fact, there is a striking difference. The experimental curves in fig. 10 lie practically parallel to one another at the higher intensities, and the point of in- flexion (which corresponds to the maximum ordinate in the nuclei distribution curves in fig. 11) moves towards the origin as the grain size increases. In curves (b), (c), and (d), fig. 10, which are for exceedingly sensitive grains, the point Characteristic Curve of a Photographic Emulsion. 367 of inflexion has moved so near the origin that the part of the curve to the left of this point does not show on the scale to which the curves are plotted. On the other hand, the theoretical curves in fig. 12 are characterized by the fact Fig. 12. that the ratio of the ordinates for different sizes of grain is independent of the intensity, and the value of I at the points of inflexion, 1; and the average sensitivity do not change as the grain size is varied. Thus we cannot explain the effect of a variation of grain size on Svedberg's assumption. Xow let us assume that the sensitivity of a nucleus depends on the size of the grain in which it is contained, and that if Fig-. 13. it is in a large grain it is more sensitive than it would have been in a small one. The effect of this on the distribution curve for the larger grain in fig. 11 is to shift it bodily nearer the zero, thus decreasing the value of I; and increasing the average sensitivity, as in fig. 13. The N I curves 368 Mr. F. C. Toy on the Theory of the plotted from these distribution curves are shown in fig. 14, and it will be seen that they are similar to the experimental curves in fig. 10. The reason why, for very sensitive grains, the lower half of the 8-shaped curve appears to vanish (b), (c), and (d), fig. 10, is that the value of Ij is very nearly zero, but it would be shown if the points were plotted on a bigger scale. Fig. 14. The evidence thus points to there being two reasons why large grains are more sensitive than small ones. Firstly, there are more nuclei present in the larger grains, so that a single grain has a greater chance of having at least one ; and secondly, the average sensitivity of the nuclei increases with the size of grain. Svedberg in his most recent paper (ibid.) discusses the relation between the average number of nuclei per grain and the grain size. He says: — " The rapidity of the increase of the average number of nuclei per grain N with size of grain would depend on two factors : the ability of the developer to penetrate into the grain, and the homogeneity of the field of light in the grain. If the developer is not able to get into the interior of the grain, but only attacks the surface layer, then N would mean the number of centres in that surface layer, and therefore would increase in approximate proportion to the grain surface even in cases where the field of light in the grain was not homogeneous (because of strong- light absorption). On the other hand, if the developer is to penetrate the grain, N~ would depend upon the field of light in the grain. If the absorption of light were feeble, N would increase in proportion to the volume of the grain ; if the absorption were very strong, N would increase approximately proportionally to the cross-section of the grain/' Later in Characteristic Curve of a Plwtograpliic Emulsion. 360 the paper he compares the variation of N with grain size for grains which have been exposed to light with the variation when the exposure is to X-rays, and suggests certain deduc- tions as regards the absorption of light and X-rays by the silver halide from the difference which he finds. Now, from fig. 10 we see (since A=N ) that the manner in which N varies with grain size depends on the intensity to which the grains have been exposed ; we can select an intensity such that N varies in almost any manner we please. Thus, unless the difference between Sved berg's results and those found here is due to the different emulsion used, there seems to be no justification for making deductions from the relation which is found between N and the size of grain at one fixed arbitrary exposure. The theory which has been advanced here is capable of explaining an important fact which appears quite inexplicable on such a theory as Allen's (ibid.). It is well known that the sensitivities of the grains in an emulsion depend to a great extent on the conditions of precipitation and ripening; and that, in different emulsions, sets of equal-sized grains may have quite different sensitivities, and even different maximum slopes for their characteristic curves. If, as Allen suggests, the nucleus is really a simple molecule of silver halide which has lost an electron, its characteristics will be the same whatever the emulsion, and it is difficult to see why grains in one emulsion should be more sensitive than those of the same size in any other emulsion. If, however, the nucleus is not silver halide, it is very probable that the conditions of pre- cipitation and ripening do play an important part in deter- mining its characteristics. Thus, on Ren wick's theory, the condition of the colloidal silver which is produced will certainly depend on such factors as the kind of gelatin, conditions and time of ripening, etc., and the ease with which colloidal silver particles can be coagulated will be affected by the amount of gelatin present, since this is a protective colloid. The great difficulty in accepting Renwick's theory as it stands is this : — It is known that an unprotected silver sol is very stable to the action of light. Therefore, if a protective colloid is present, it will be still more difficult to effect its coagulation and precipitation by light, whereas in the case of our most sensitive silver halide grains the energy necessary to make them developable is exceedingly small. Luppo-Cramer (ibid.) believes that the mechanism of the formation of the latent image is not the same for the most sensitive and very insensitive emulsions, and he claims that Phil. Mag. S. 6. Vol. 44. No. 260. Aug. 1922. 2 B 370 Characteristic Curve of a Photographic Emulsion. this is supported by his experiments. He found that the sensitivity of a very fast emulsion was decreased considerably by treatment with chromic acid, but that the sensitivity of a very slow emulsion remained unchanged. He explained this by the existence on the surface of the sensitive grains of colloidal silver, formed during the ripening process, which was not present in the insensitive grains, and which was removed by the chromic acid. It is very difficult to imagine that the fundamental light action varies with the kind of emulsion, and that considering a whole series of emulsions, from the most sensitive to the most insensitive, there is a transition region where an entire change of mechanism takes place* Strong evidence against Liippo-Cramer\s view is that Svedberg (ibid.) has shown that in one of the slowest emulsions the reduction centres are distributed amongst the different grains according to the same law as has been shown here to hold for their distribution in the case of one of the fastest commercial emulsions. This is in favour of the view that for all kinds of emulsions the process of the formation of the latent image is the same. The existence of this chance distribution of developable " centres " in the grains does not conclusively prove that they are the kind we have considered in this paper, and there are at least three other possibilities. Assuming a discrete structure of the radiation, the centres may, as suggested by some, be the points of impact of light quanta on the grains, but the fact that the majority of these centres are located on the edges of the grain is strongly against this view. Also within the crystal there maybe a chance concentration of the light energy at certain points, and both these possibilities are being tested in this laboratory. Again, this chance dis- tribution may be due merely to the fact that the grain as a whole is changed by the light, but the developer reaches some points of it sooner than others. If this is so, there appears to be no reason why the average number of centres per grain, considering only developable grains, should in- increase, as it does, in a regular manner with the light intensity. The author believes that the evidence so far obtained is mainly in support of the theory discussed in this paper. In conclusion, the author wishes to express his thanks to Dr. T. Slater Price, Director of Research of the British Photographic Research Association for much valuable criticism and advice. . On the Stark Effect for Strong Electric Fields. 371 Summary. A theory is advanced which explains the relation found experimentally between the number of geometrically identical silver halide grains made developable and the light in- tensity. It is assumed that there exist in the grains particles which are not silver halide, and which are formed during precipitation and subsequent ripening. With any normal exposure (i. e. one which gives a value between and 100 for the percentage of developable grains), it is these particles which form the reduction nuclei, the only action of the light being to change their condition in such a way that they become susceptible to the action of the developer. Each nucleus does not necessarily require the same intensity to change it. The nuclei are scattered haphazard amongst the grains according to the laws of chance, and only grains which have at least one will be developable. The sensitivity of a grain is the sensitivity of its most sensitive nucleus. The effect of a variation of grain size is explained, and it is shown that Svedberg's assumption regardiDg the similaritv of the light-sensitive material in large and small grains is not in agreement with the experimental facts in the case of a fast emulsion. M XXXII. On the Stark Ejfeet for Strong Electric Fields. To the Editors of the Philosophical Magazine. Gentlemen, — Y attention has been drawn to the results of experi- ments by Takamine and Kokubu * in which an effect of the nature indicated in a recently published paper t of mine was detected, namely, a shift of the central line in the perpendicular component of Hy in a strong electric field. Before comparing the experimental amount of this shift with the theoretical value it would have on the Quantum theory of spectral lines,, it is necessary, however, to point out a slip in my paper referred to above : thus on p. 945 a term is missing from the value of: the contour integral (4), instead of (6) the full value should be f = B D /3B* \ 5BD*/ 7B»\^ * "The Effect of an Electric Field on the Spectrum Lines of Hydrogen," I'art III. Memoirs of the College of Science, Kyoto Imperial University, vol. iii. p. 271 (1919). t Phil. Mag. xliii. May 1922, p. 943 ; this will be referred to freely. 2 B 2 372 On the Stark Effect for Strong Electric Fields. Consequently the third term on the right-hand side of equation (2b) p. 948 should be where N' is now given by -15n 3 2 -21(n 2 -n 1 ) 2 ~6-—^ ) i -- L 3 v 2 iy nx + wa + rjj, j N(n) being still given by equation (10). In view of the identity N(n) = (2n 1 4-^3)(6n 2 2 4-6n^ 3 + n 3 2 ) + (2?2 2 + W 3 ) (6?2 X 2 + 6 WjWg + Tig 2 ) = 3(n x + w 2 + ?2 3 ) 3 — 3(ni 4- n 2 + w 3 )(n 2 — n x ) 2 — « 3 2 ( w i + w 2 + w 3 ), N' can be reduced to the form N'(n) = (n 1 + n 2 + n 3 ) 4 {17(n 1 + n 2 + w 3 ) 2 --9ri3 2 ---3(n 2 - y i 1 ) 2 } 5 ... (ii.) which shows in conjunction with (i.) that the remarks in the paper about the symmetry of the components are not affected by this correction. In order to calculate the amount of shift of the middle w-component of H y we observe that this component can arise from any of three possible transi- tions corresponding to (m 3 . m 2 , wii ; n 3 . n 2 , n ± ) = (3, 1, 1 ; 2, 0, 0) or (1, 2, 2 ; 2, 0, 0) or (1, 2, 2 ; 0, 1, 1) respectively, the values of {N'(n) — tN'(/n)} corresponding to these combinations being — 2'MxlO 5 , — 2*59 x 10 5 7 and again — 2*59 x 10 5 respectively. And on substituting the values of the universal constants in (i.) for hydrogen (E = e) the following expression is obtained for the wave-length shift \ 2 \ 2 F 2 Damping Coefficients of Electric Circuits. 373 This gives for H y (X = 4'34 x 10~ 5 ) and the value of F used by Takamine and Kokubu *, namely, F = 4-33 x 10 2 c.g.s. e.s. units [ = l'3x 10 5 volt x cm.- 1 ], AX, = *36A or *43A respectively. The experimental value observed by Takamine and Kokubu is about 1A, which is larger than that predicted by the Quantum theory. It is, however, possible that part of the experimental shift is due to a Doppler effect, and in any case the experiments could hardly be considered accurate enough to exclude o a possible experimental error of what is only about ^ an Angstrom unit. On the other hand, the photographs of the shift [plate ii. fig. 1] point decidedly to a (general displacement of all the components in the direction pre- dicted by the theory, namely towards the red, and may be taken as corroborative of at least this qualitative aspect of it. It is seen from (i.) and (ii.) that this lack of symmetry in a strong field would be expected on theoretical grounds to be more pronounced for the higher members of the Balmer Series {e.g. H$ or H 6 ), and it would be highly desirable to obtain measurements relating to these lines as a further test for the quantitative aspect of the theory. In conclusion I wish to thank Mr. W. E. Curtis for drawing my attention to the experimental results already referred to. T ^. , nil T q Yours faithfully, Kings College, London, J ' May 12th, 1922. A. M. MOSHARRAFA. XXXIII. On the Damping Coefficients of the Oscillations in Three- Coupled Electric Circuits. By E. Takagishi, Electro -technical Laboratory, Department of Communica- tions, Tokyo, Japan f. THOUGH the importance of the problem of three-coupled electric circuits has arisen with reference to radio- telegraphy, it does not seem to have been attacked with any great amount of attention except by B. Macku, E. Bellini, and very recently L. C.Jackson \. The valuable paper of the latter made me feel very much interested, especially as it will make an important contribution to radio fields, but * I. c. f Communicated bv the Author. X Phil. Mag. vol. xlii. No. 247, July 1921, p. 35. 374 Mr. E. Takagishi on Damping Coefficients of unfortunately I found there slight errors concerning the damping coefficients of the circuits. Now, let us proceed to correct them, using the same notation and abbreviations for the sake of simplicity. Comparing coefficients in the . equations (5) and (6) in his original paper, we obtain, instead of (7), -2(q + r + s) = (R-iL 2 L 3 H- L 1 R 2 L 3 + L 1 L 2 R-3 -(co^ + co^ + co/) _ J L^-M^ 2 L 2 L 3 -M 23 2 L 3 L 1 -M 31 2 + L 1 R 2 R 3 +RiL 2 R3 + R J R 2 L 3 \-r-D, . (ii.) -2{co 1 2 (r + s) + co 2 2 (s + q)+co 3 2 (q + r)} RJjg + LiR-a . R0X3 + L 2 R 3 R^ + LA C ^ d + ~cT + R 1 R 2 R 3 |--D, . (iii.) C 2 C 3 C1O3 CiG 2 RiR 2 R 2 R 3 R 3 Ri\ . -n r * 2(&) 1 2 ft} 2 2 5-f- co 2 2 co 3 2 q -f cwg 2 ©! 2 **) •«i 2 (»2 2 a>3 2 QAC -e-i), (tL; (7') where D = 2M 12 M 23 M 81 -L 1 L 2 L 3 + L 1 M 23 2 +L 2 M 31 2 + L 3 M 12 2 . (vii.) From equations (7') ii., iv., vi., vii. we obtain the same equation for co 2 as (8) and (9) in his original paper. For the damping coefficients q, r, and s, however, we find the following values, different from, those in (10). Oscillations in T 'hree- Coupled Electric Circuits. 375 Now, making use of the abbreviations given below and in the original paper, we get the following equation : -2(?+ ;• + .<)=>, - 2 {./(or./ + O + K» 3 2 + °>i 2 ) + «(»i a + » 2 2 ) ! ■ = | , 2f V 9 , 9 9. 9 &)o-.^ = c X where A = * 1 (l-/9»)+*,(l-7»)-f*,(l -a 8 }, C = k^irn' 2 -{- Ii\ 2 P)i 2 + kj 2 m 2 , X = -(l-a 2 -/S 2 - 7 2 -2a^ 7 j. Solving these simultaneous equations, we have Q R '' = Y' S S =Y' in which A X Q = — 2 v- — 2(w 3 2 + w 1 2 ) — 2(oj 1 2 + &) 2 2 ) C X !(&) 3 2 o) 1 2 ) — 2(o) 1 2 6) 2 2 ) A 1 1 B (a)a 2 + ^ 2 ) K 2 + r, 2 2 ) /" ^ 9 9 2 9 = |.(co 2 2 -a) 3 2 ) {A<-B Wl 2 + C}, similarly R = i (co 3 2 - Wl 2 ) {4« 2 4 -Bw 2 2 + C}, and Y - 8 (a>, 2 -G) 2 2 ) (of—uf) (a), 2 - *»/) 376 Prof. S. C. Kar on the Eleetrodynamic That is, q = [^ 1 mV + ^n 2 /H/^H 2 )-o) 1 2 {/ 2 (/c 2 + ^) + m 2 (A-3 + ^ 1 ) 4-n 2 (^ 1 -+^)+A ;i /cA} + a) 1 4 {^(l-/3 2 )+^(l- 7 2 ) + /, 3 (1-^)}] - : - 2 (l_ a 2_£2_ 7 2_ 2a/ 5 7 ) (»*-**) (tof-CDs 2 ), r = [(/^ 2 +/^ 2 + y 2 Hi 2 )- w 2 2 { P fe + y -\-m 2 (h + h) + n 2 ft^/. 2 ) + /^ 2 /^l + a, 2 4 {^(l-/3 2 )+A- 2 (l- 7 2 ) + hil -a 2 )}] ^2(l-a 2 -6 2 -7 2 -2a y S 7 )(a) 2 2 -a) 3 2 )(co 2 2 -a) 1 2 ), + 72 2 ft + /v 2 ) + Ws} + o) 3 4 {^(l-^ 2 ) +/c 2 (l- 7 2 ) + ^3(l-« 2 )}] ^2(l-a 2 -/3 2 - 7 2 -2^ 7 ) (» 3 2 -ft>i 2 ) (a) 3 2 -0. On inspecting these equations for damping coefficients it is noticed, at once, they are also correct with respect to the dimensions. XXXIV. On the Eleetrodynamic Potentials of Moving Charges. By S. C. Kar, M.A., Professor of Mathe- matics, JBangabasi College, Calcutta *. THE eleetrodynamic potentials of a moving charge or the electron have been the subject of several in- vestigations and the earliest were those of Lienardf and WiechertJ. Among recent writers who have found the potentials on a relativity basis may be named Sommerfeld§ and M. X. Saha||. Both of these writers performed a four- dimensional integration in the Minkowski space-time mani- fold and have obtained results which are quite general. It appears to the present writer that the Lienard and Wie- chert result — and the method admits of easy extension to the case of a straight linear current — may be obtained easily enough by a Lorentz transformation to a rest-system and back without resort being had to four-dimensional integration. * Communicated by the Author. f JJ Eelairage electrique, vol. xvi. pp. 5, 53, 106 (1898). t Arch. Need. vol. v. p. 549 (1900). § Ann. (I. Phys. vols. Hi. and liii. j| Phil. Mag. vol. xxxvii. p. 347 (1919). Potentials of Moving Charges. 377 The equations for the potentials may be written □ (ic<&, F, G, H) = /3 /c(zc, u, r, ic) \ i / A a 2 + i' 2 H-«> 2 where /c=1/a/ 1— 2 It is well to point out that this mode of writing the equations is slightly different from the customary mode where QF-M so that our F is d¥'. This deviation c from usage is justified by the greater symmetry and homo- geneity of form resulting. The equations for h (magnetic intensity) and d (electric intensity) will on account of this •change assume the forms h=- rot(F, G, H), (2) It is evident that the operator □ is an invariant under n Lorentz transformation. It will therefore follow that (ic<&, F, Gr, H) is a four-vector, because ic(ic, u, v, iv) is a four-vector. Therefore c 2< &8t — FSx — G8y — HSr which represents the scalar product of the four-vectors (zc<l>, F, G,H) and (ic8t } 8x. hy, 8z) is invariant under a Lorentz trans- formation. Therefore, e 2 <P8t-F8x-G8y^R8z=c 2 &8t , --F'8x'--G'8y'--K'8z', where the dashes refer to a system of axes moving with velocity v along the axis of x. But 8x= K (8x' + v8t'), Sy = Sy, Sz = 8z', and bt = K ht' + =\ where k = l/.y/ 1 - % . Substituting and equating coefficients of 8x', 8y', Sz', and ■8t' we have F'= K (F-v®), G' = G, H'=H, F and <&' = *(<I> — ^-Y 378 Prof. S. C. Kar on the Electrodynamic These formulae are exactly similar to the usual formulae for Sx, 8y, Sz, 8t and connect the potentials for any system of axes with those of another moving with velocity v along the axis of x. The reversing formulse are F = *(F' + v<S>'), G = G', H = H', and ^> = /e /V+^. (3) Let us suppose an electron moving with velocity v along the axis of x and let us take a system of axes moving with the electron. It is apparent that for the latter system of axes the electron is at rest. The vector potentials (F', G', H') = and <!>' = - — , due to a static charge e where r' is the distance of the point P at which the potentials are considered. For the original system of axes, therefore, we should have according to the formulae of transformation given above, F=/™ * , G = 0, H = 0, and <£ = *«,—,. t is, however, expressed in terms of the coordinates of the rest system and it will be necessary to transform it to a form involving the coordinates of the original system. But if the time-difference between the point P and the electron is A*', then r' = cAt' - KG \ At r vAx~] r\ v Ax~\ — K\r = kt 1 . — L c J L c r J ev F = -,G = 0, H = 0, and <E> — 4:TT7 I" 1 --]' which are Lienard's results. Potentials of Moving Charges. 379' (4) Lei us suppose a straight linear uniform current to arise from continuous and uniform rush of electrons in the conducting wire in the direction of the current. Viewed from a system of axes moving with the common velocity of the electrons the phenomena reduce, as far as the rushing electrons are concerned, to the case of a linear and uniform distribution of electric charge. If N electrons each with charge — e be supposed to rush with velocity v to the observer in the rest-system the linear density of static charge is — N*. From the ordinary theory of potential, the potential <E>' for such a distribution is — 2N<? logV where r l2 =y' 2 + z' 2 and (F', G', H') = 0. Transforming to a moving system according to our formula? we should have F = Kv&=-2/eNev\ogr, G=Gr'=0, H = H' = 0, <5? = K & = -2/cNelogr [v r-=r']. The magnetic field therefore would be given by h x = 0, 2xNevz 1 2/eNm/ , , . . „ , n v = 5 — , h = ^ —■ and the electric held would be , , A 7 2/cN^y , 2/cNez T ,, given by d x — 0, Oy^= y^ , a*= ^— . In the con- ducting wire, however, there is also a linear distribution of positive nuclei at rest of which the potential would be + 2N<? log r. The electric field due to these would be given by d/ = 0, , mey ■ , M 2-Nez The resultant electric field would therefore have the components 0, !L ~V (1— /e), - — ~ (1 — k), and is of the order v 2 of 2 . The magnetic field is of finite magnitude and cir- cular round the wire, the resultant being which is ., . , . , . . 2 (current) .„ quite in accord with the expression it we put k±s ev iS ev current = or neglecting quantities of the order v 2 2 in comparison with unity. [ 380 ] XXXY. The Identical Relations in Einstein' 's Theory. By A. E. Harward*. THE March number of the Philosophical Magazine con- tains an interesting proof of the identity by Dr. G. B. Jeffery. Apparently it is not generally known that this identity is & special case of a more general theorem which can be very easily proved. I discovered the general theorem for myself, but I can hardly believe that it has not been discovered before. The theorem is (BpyoPjr + (B^/)„ + (BptS)* =0. . . (1) This identity can be verified in a rather laborious manner by forming the covariant derivative of B^^P, but it can be more easily proved as follows : — The identity Aju, V( j — Aju, <Tv — j5fivo p Ap .... (2 J can be easily generalized so as to apply to the case where instead of the vector A^ we have a tensor of any order ; thus Aftv, or — Apy, tg == -t>ju<7r Apv ~T Dyur -^H-P' This is proved in the same way as (2). Now, if Afx be any covariant vector, then (Aju, vot — Aju, vt(t) -f (Aju, arv — &p., <rvr) + (^/z, rva A^ ; rav) = (A^, va — A^ ; cv)t-{- (A^, ar — Ap, rtrjv + (A^, rv — A^ V r)a\ J3 JLt(Tr |0 A.p^ v -\- D V(TT P Ap ( p -4 lJfiTV ^-p, <T + *5<TTV Ayu, p + ^fxva P Ap, T + ¥>TV(T P Ap 5 p = (Bp Va P Ap)r + ( B^ T P A p ) v + (B Mr / Ap) a . * Communicated by the Author. The Identical Relations in Einstein s Theory, 381 Now, (B^/ A p ) r = (IV/) r A p + BpvS A Pl r ; so after cancellation we get (B vff r p + B ( rr/4-Br V /)A /t p The expression in brackets on the left vanishes identically. Since Ap is arbitrary, the expression in brackets on the right must also vanish. Q.E.D. The identity B vaT P + B aTV P + B TV a<> = follows at once from the well-known identical relations between the Riemann symbols. The three-term identity is usually stated in the form (/uLTav) -f (/jlcvt) -f (/jlvtct) = 0, or in the modern notation Bfxpcrr~\~ ±>[xtv(t ~\~ tjjxarv == 5 here B^cr denotes greB txva 6 = (firav), Since B [irv(T = 'B 1/(Tf j ir . and Bh(ttv = BcffivTi = B pV (JT 4" Bpa/JLT T j5<T/J.V7 We assume that the determinant \g^v | = g does not vanish in the region under consideration ; therefore the expression in brackets must vanish. This identity can also be proved by observing that the expression (Av, or — Aj^, to) + (Aff ) tv — -A-ff, vt) 4" (A r, vff A-t, <jv) vanishes if A v is the derivative of a scalar; for in that case A Vt crr — A.cr,vT, Aa t Tv == At, <tv> and A^yu = Aj/,rcr. If we contract (1) by putting r = p, we get 382 Mr. H. S. Rowell on Energy Partition If we contract this again by multiplying by gP v , we get the familiar identity <-H-?- ' (4) for since [gP v )p = 0, similarly and *r <f (G r , ff ) = (/» G„„)„ = (G)„ =g since G is a scalar. Jersey, 13th May, 1922. XXXVI. Energy Partition in the Double Pendulum. By H. S. Rowell *. IN a letter to ' Nature ' (July 28, 1921) the present writer gave a theorem on the double pendulum which is capable of interesting extension. If the masses of the bobs are m and M and the respective amplitudes are a and A with suffixes. to denote the normal modes, then the theorem states that a Y a 2 M AjA 2 m ' If this equation is squared and both sides multiplied by m 2 /M 2 , we have ma^n 2 ma 2 2 n 2 2 _ 1 — 1» MAxVMA/tis where nj and n 2 are the radian frequencies of the two moles. "This equation may be readily interpreted thus : — " The ratio of the kinetic energy of one bob to that of the * Communicated by the Author. in the Double Pendulum. 383 other bob in one mode is the reciprocal of the corresponding ratio in the other mode/' Proceeding to the general case of an elastic system with two degrees of freedom, using Professor Lamb's notation, 2T = A6 2 + 2U0cf> + B4>\ 2 V = 2V + ad' 2 + 2h6<f> + b(f> 2 j so that with a time factor = in 7i-(A0+H<£) = a6 + l«f>, ?r(H<9 + B(£) = 7,0 + ty; whence the product of the roots in 6j^> is Ti — / & • If H = so that T is a function of squares of velocities, the product of the amplitude ratios is — B/A, or, in the double pendulum, — M/m. If A = so that the potential energy is a function of squares of displacements, the product of the amplitude ratios is bja, i. e. the ratio of the two stability coefficients. Thus in either case we have an energy relation. For the kinetic energy take H = and ££=-?; 9192 A •square and multiply by A 2 /B 2 , and insert the frequencies. For the potential energy take A = and -V = — - 0102 i 2 <* which, when squared as before, yields a similar relation. The two results may be expressed in words thus : — When the Kinetic or Potential Energy is written as a ■function of squares only, the ratio of the Kinetic or Potential Energy expressed in one co-ordinate to that •expressed in the other co-ordinate for one normal mode is the reciprocal of the corresponding ratio for the other normal mode. This investigation gives an insight in certain cases into the indeterminateness of the normal modes with equal periods. [ 384 ] XXXVII. Velocity of Electrons in Gases. To the Editors of the Philosophical Magazine, Gentlemen, — IN a paper in the Jalirbuch der lladioactivitCd und Electronik (vol. xviii. p. 201, April 1922) H. F. Mayer gives an account of some of the formula? obtained by different physicists for the velocity of ions or electrons in gases due to an electric force, and concludes that a formula recently given by Lenard is more correct than the others. Among the other formulae which are discussed, the author gives what purports to be an account of a formula for the velocity of an ion which I published in the ' Proceedings of the Royal Society' (A. vol. lxxxvi. p. 197, 1912), and states that this formula is so incorrect that it does not even give the right order of the velocity. I should like to draw attention to the way in which Mayer has misinterpreted the matter, and to quote the formulae as I gave them for the different cases in which the mass of the ion is small or large compared with the mass of a molecule of the gas through which it moves. On pages 199, 204, and 206 of my paper, three formula? are given for the velocity U of an ion in the direction of the electric force X in terms of the mean free path I of the ion, its mass m, charge <?, and velocity of agitation u which is supposed to be uniform and large compared with U. The first of these is JJ = Xel/mu, (1) and applies only to cases in which the mass of the ion is small compared with that of a molecule of the gas (an electron for example), since it is here assumed that after a collision with a molecule all directions of motion of the ion are equally probable. I pointed out that when the mass of the ion is larger than that of a molecule of the gas, all directions of motion of the ion after a collision are not equally probable, and that in this case an ion travels a considerable distance (having an average value X) after a collision in the direction in which it was moving before a collision. A more general formula for the velocity was given, which is XJ = Xe(l+X)/mu. . . ! . . (2) If the mass m of the ion is so large compared with the Velocity of Electrons in Gases. 385 mass m of a molecule of the gas that all directions of motion of a molecule become equally probable after a collision with an ion, it was shown that formula (2) reduces to U=Xel/m'u, (3) as in this case it may be seen that l + X m m (4) It will be observed that formula (2) reduces to (1) when A, is zero, that is when m is small compared with m\ so that either of these two formulae may be applied to the case of an electron moving in a gas. Mayer, however, selects formula (3) to find the velocity of ions of small mass or electrons, although it is definitely stated in my paper that formula (3) refers to large ions, and the relation (4) on which it depends can only hold when m is greater than m! '. As the correct formula (1; for electrons differs by the factor m'/m from formula (3), it is unreasonable to expect the latter formula to give the velocity of an electron. The above formulae, obtained by simple considerations when the velocities of agitation of all the ions are taken as being the same, are of course not absolutely exact. There is a numerical factor by which the expressions should be multiplied in order to allow for the variations of the velocity of agitation about the mean velocity. In the most interesting case, which is that of electrons moving in a uniform electric field, the value of the numerical factor is about '9, but it has not been determined exactly. The determination of this factor is very difficult, as the distribution of the velocities of agitation of the electrons depends on the energy of an electron which is lost in a collision, and experiments show that the proportion of the total energy of an electron which is thus lost depends on the velocity. This problem has been fully considered by F. B. Pidduck (Proceedings of the London Mathematical Society, ser. 2, vol. xv. pt. 2, 1915), who shows that under certain conditions the proportion of the velocities which differ largely from the mean velocity of agitation is much less than the proportion indicated by Maxwell's formula for the distribution. It appears that the error introduced by taking the velocities of agitation as being all equal to the mean velocity may be no greater than when the velocity distribution is taken as being the same as that given by Maxwell's formula. [n order to obtain an exact formula for the velocity U it Phil. Mag. S. 6. Vol. 44. No. 260. Aug. 1922. 2 386 Prof. H. A. McTaggart on the Electrification would be necessary to take into consideration the variation of the mean free path of an electron with its velocity of agitation, and the large reduction of the energy of an electron when ionization by collision takes place. These points in connexion with the motion of electrons in gases have not been taken into consideration by Lenard, and it does not appear that his formula is more correct than others which have been proposed. Yours faithfully, 3rd May, 1922. JOHN S. ToWNSEND. XXXVIII. On the Electrification at the Boundary between a Liquid and a Gas. By Professor H. A. McTaggart, M.A., University of Toronto *. MANY years ago, in the course of some experiments on the effect of an electric current on the motion of small particles in a liquid, Quincke {Ann. d. Phys. cxiii. p. 513, 1861) observed that small gas-bubbles in water moved as though negatively charged. Although a good deal of atten- tion has been paid to the movement of solid and of liquid particles in such cases, very little effort has been devoted to the study of small spheres of gas suspended in a liquid — one obvious reason being the difficulty of controlling them while under observation. A systematic examination of their elec- trical properties ought, however, to yield further information as to the physics — and chemistry too — of surface layers. Before the war experiments in this field were begun by the author in the Cavendish Laboratory under Sir J. J. Thomson, and some results were obtained. Measurements were made ^Phil. Mag. Feb. 1914, p. 297) of the velocity, under a fall of potential, of small spheres of air in distilled water and their electrical charges were estimated. The effects on the charge of the addition of minute amounts of various inorganic electrolytes were studied. Results were obtained (Phil. Mag. Sept. 1914, p. 367) showing how the charge varies with the presence in the water of certain alcohols and organic acids, and a parallel was shown to exist between the variation of the electric charge and the surface tension. The present paper deals with some further experiments carried out in the University of Toronto, and describes the variation observed in the electric charge on small spheres * Communicated by Professor J. C. McLennan, F.R.S. at the Boundary between a Liquid and a Gas. 387 of air when a particular electrolyte, Thorium Nitrate [Th(N0 3 )J, was dissolved in water. This salt was selected tor special study because it had been found to be unusually active in charging these surface layers. The apparatus used was similar to that referred to in a former paper, one or two changes being made in it for greater convenience. The arrangement is shown in fig. 1. Fig. 1. K - T BT J OlD A is a small cylindrical glass cell rotating about its axis on pivots and driven by a belt of thread from a pulley F on a Rayleigh motor. This motor was made in the laboratory workshop, and has, instead of the usual fly-wheel with a hollow rim filled with water, a solid brass wheel H — a modi- fication suggested by Professor Wilberforce of Liverpool. The wheel, although loose on the shaft, has enough friction, when a heavy oil is used for lubricant, to keep the shaft in steady motion after synchronism with the tuning-fork is attained. D is a timing device consisting of a vertical post carrying a pointer and made to rotate by a toothed wheel working in 9 Q v 388 Prof. H. A. McTaggart on the Electrification the worm E. The pointer rests by its own weight on the top of the post, but at any instant in its motion over the fixed dial D it may be raised and stopped by a small electromagnet controlled by the key B. When released it falls back on the post and begins to record time with the same regularity as the tuning-fork. It forms a very convenient stop-watch if velocities are to be measured. A travelling microscope M measures the distance travelled by any bubble on the axis of the rotating cell. The water used was twice distilled — the second time in " Pyrex " glass and condensed in a silver coil. The thorium nitrate was by Merck, and was assumed to have 12 H 2 — water of crystallization. A stock solution was made up containing 4 x 10~ 6 equiva- lents per c.c. (1/250 normal), and from this other solutions were made by successive dilation. A first series of readings was taken with various concen- trations of the salt, but with bubbles of nearly the same size in order to reproduce the effects previously observed — the method of working being to fill the cell A with the desired solution, introduce a single bubble of air with the gas pipette, and set the cell in rotation. The bubble very soon takes up a steady position on the axis, and its motion under any fall of potential L may be examined. Very small concentrations sufficed to reduce to zero the natural negative charge found in pure water and to give the small sphere of air a positive charge. The following readings are typical : — Fall of potential 34 volts per cm. Diameter of bubble 0*3 mm. Concentration. ^. f Velocity of Equivalents ^ n ot bubble. per c.c. c lar S e - cms./sec/volt./cm. 4X10~ 7 + 5xl0~ 4 4X10-8 m . „ + slower 8X10" 9 + very slow (1-5X10- 4 ) 5-7X10'" 9 - slow 47X10- 9 - faster 4xl0 -9 - faster Pure water — 4xl0 -4 at the Boundary between a Liquid and a Gas. i^89 The zero point was reached at a concentration o£ about 7 x 10~ 8 , a result rather higher than that given in a former [>aper. The salt was an entirely different sample, and may not have contained the same proportion of water of crystal- lization. (See Abegg and Auerbach, ' Inorganic Chemistry.') A series of readings was then taken for spheres of air of different sizes, one object being to observe the charge on very small spheres. It is very difficult, by the use of any kind of pipette, to introduce into the rotating cell bubbles smaller than 1/5 mm. in diameter. To avoid this difficulty the following mode of working was adopted. The solution was first placed in a partial vacuum to remove as much dissolved air as possible, and afterwards poured into the cell. A bubble into this gas-free solution slowly decreased in size by absorption until it vanished, while the electric charge could be observed at any stage. Under these circumstances it was found that for a suitable concentration of solution a sphere of air which began with a small negative charge almost invariably and in a regular way reduced its charge to zero, and gradually took on a positive charge. The following readings illustrate this point — No. 1. Concentration. Equivalents per c.c. ~ 9 Xo-7 Diameter of sphere in mm. 0-26 0-17 Sign chare - 0-14 - 0-10 0-08 + No. 2, 10- Concentration. Equivalents per c.c. X5*7 Diameter of sphere in mm. 0-44 0-35 0-26 0-17 0-14 Sign of charge. 390 Prof. H. A. McTaggart on the Electrification No. 3. Sign of charge. Concentration. Equivalents per c.c. i~ 9 X5-7 Diameter of sphere in mm. 0-62 0-53 0-39 0-17 0-08 No. 4. Concentration. Equivalents per c.c. Diameter of sphere in mm. Sign charg »~ 9 X5-7 0-71 0-53 — 0-44 — 0-35 0-32 + 0-23 + Concentration. Equivalents per c.c. 10- 9 x5-7 No. 5. Diameter of sphere in mm. 0-28 0-17 O08 005 Sign of charge. It will be seen from the first four examples given that at a concentration of 10~ 9 x 5*7 the change of sign occurs in every case. Rarely, as in No. 5, and then only when the original sphere was small, did the sign remain the same. Even then the charge grew steadily less. In practically every case the negative charge slowly decreases as the bubble gets smaller, passes through zero, and increases to a small positive value. at the Boundary between a Liquid and a Gas. 39 1 Three examples are given for slightly greater concentra- tions : — No. 6. Concentration. Equivalents per c.c. Diameter of sphere in mm. Sign charo 10- 9 x6-6 0-53 041 0-35 + 0-17 + 0-08 + No. 7. Concentration. Equivalents per c.c. 10" 9 X6-6 Diameter of sphere in mm. 0-35 0-28 0-26 0-17 0-14 Sign of charge. Concentration. Equivalents per c.c. 10" 9 x8 No. 8. Diameter of sphere in mm. 0-44 035 017 Sign of charge. Almost zero. Above a concentration of 10~ 9 X 8 the bubbles were always positive. The examples given show that the spheres do not all have the same size when they reach the zero — isoelectric — point in a given solution. The larger a sphere is at the beginning the larger it is when its charge becomes zero. This suggests, as the cause of the change in sign, a kind of coagulation of something in the free surface. It is known that, in a solution of thorium nitrate in water, 392 Prof. H. A. McTaggart on the Electrification hydrolysis occurs with the formation of thorium hydroxide thus — Th(N0 3 ) 4 + 4 HOH— >Th(OH) 4 + 4 HN0 3 . There is present in the solution some of the original salt, some acid, and the hj-droxide in colloidal form. The pre- sence of the last-mentioned was suspected as one of the causes producing the reversal of sign, and experiments were then made to test its activity in altering the charge. A colloidal solution of thorium hydroxide as free as possible from salt and acid was prepared by dialysis (Burton, 4 Physical Properties of Colloidal Solutions,' 2nd Ed., p. 16). A dialysing " sleeve " shaped in the form of a test-tube was made of " parlodion " (sold by the Du Pont Chemical Co., New York). A solution of the parlodion in ether and alcohol was used to coat the inside of a test-tube of suitable size. After the solvent had evaporated the parlodion re- mained as a thin but strong film which when detached from the glass served very well as a dialysing vessel. For this experiment a solution containing about 2 gm. of salt in 50 c.c. of water was dialysed for a period of three weeks, after which an estimate was made of the colloid pre- sent. A sample of 10 c.c. evaporated over sulphuric acid gave a residue of '0034 gm. The residue formed a thin layer of gelatinous material on the bottom of the evaporating dish, with drying cracks across it in all directions. The effect of this colloid on the charge on small spheres of air in water was then examined, the dialysed solution above mentioned being diluted as shown in the following examples : — No. C.c. colloid solution in 100 c c. water. Diameter of sphere in mm. Sign char 1 10 0-21 + 0-12 + 0-07 4- 2 5 017 + + 3 25 0-17 0-07 + It is seen that the surface is charged positively by the presence of very small amounts of the colloid. The following examples show the gradual reversal of the at the Boundary between a Liquid and a Gas. 393 sign of the charge accompanying the absorption of the bubble : — No. C.e. colloidal solution in 100 c.c. water. Diameter of sphere in nun. Sign of charge. 1-0 0-17 — 0-14 + 0-5 0-35 _ 0-26 — 012 + 005 + 0-25 032 - 0-17 - 0-08 — 0-05 4- The experiments show that the colloidal thorium hydroxide gives both the effects observed with the ordinary solution. It not only charges the surface positively if present in suffi- cient amount, but it also exhibits the reversal of charge with diminishing size of the bubble, and this, too, in concentrations of thorium of about the same order as in the case of the salt. Discussion. The state of the matter and the nature of the electric forces in surface layers of liquids is still a subject on which no very clear ideas exist. Experiments on electro-endosmosis all point to a selective action in such layers so far as the ions in the solution are concerned. But (he observations are always complicated by the presence in contact with the liquid surface of a solid whose role in the selecting we are ignorant of. The same is true of cataphoresis experiments with solids, as, for example, in the study of the electrical charge on colloidal particles. This difficulty is avoided, however, in similar experiments with small spheres of air — or any gas — and in such cases we can safely regard any effects observed as due largely to the properties of the liquid and its free surface. In particular, the electrical charge existing at any air-liquid surface may be considered as the result of forces residing altogether in the liquid. It ought to be possible, then, in considering potential differences at solid-liquid junctions to isolate the contribution of the liquid. In the case of thorium nitrate in solution the selective 394 Electrification at Boundary between Liquid and Gas. action of the air-water surface is very marked, a positive charge being acquired by the surface with very minute concentrations of the salt. The positive ions available for selection are Th + and H + , but neither of these separately can be responsible for the unusual activity of the salt. The mere presence of H + ions, as, for example, in the form of an acid, does not produce so great an influence on the surface charge. Nor can free Th + ions have much effect, for they disappear in the dialysis and yet leave the pure colloidal solution practically as active as before. The real agent must be the particles of colloidal thorium hydroxide which gather about them groups of H + ions and carry them into the surface in larger numbers than would be possible for the H + ions alone. The nature of this selective action must be connected with the shape of the surface, or, to put it in another way, a particle must reach a certain size before it can be regarded as having a surface-layer about it with a tension and an electric charge. We have at present in order of size — ions, ionic micelles (Prof. McBain, " Soap Solutions," Nature, March 10, 1921), ultra-microscopic colloidal particles, micro- scopic and macroscopic particles including gas-bubbles. At what stage a surface-layer is formed it is difficult to say, but it seems reasonable to suppose that the curvature of such a surface would have an effect on the charge adsorbed. The change of sign with decreasing size of air-sphere shown in these experiments seems to bear out this idea. The information obtained regarding the effect of thorium nitrate on the electrification of air-water surface layers may be summarized as follows : — 1. Thorium nitrate in aqueous solution and in concentra- tions as small as 8xl0~ 6 normal gives a positive electric charge to the surface of a sphere of air immersed in it. (In distilled water the charge is always negative.) 2. For concentrations in the neighbourhood of 6 x 10~ b normal a sphere initially negative becomes gradually positive as the sphere diminishes in size. 3. Colloidal thorium hydroxide in small concentrations of the same order also gives a positive electric charge to a sphere of air immersed in it. 4. Colloidal thorium hydroxide also exhibits the reversal of the sign of the charge with a decrease in the size of the bubble. 5. It is suggested that this reversal of sign is experimental evidence of a relation between the curvature of the surface and its adsorptive power. Lecture-Room Demonstration of Atomic Models. 395 The experiments are being- continued as time permits in the hope of obtaining* some new information regarding these free surfaces. Is is the intention to compare with thorium the effects of one or two other tetravalent and trivalent metals in the colloidal state. I wish to thank Professor J. C. McLennan for his kind and encouraoino- interest in the work. XXXIX. Note on a Lecture-Room Demonstration of Atomic Models. By Louis V. King, D.Sc, Macdonald Professor of Physics, McGill University*. [Plate II.] Section 1. SEVERAL mechanical models illustrating various types of atomic structure have been proposed from time to time. Among these we may mention Mayer's classical experiments with floating and suspended magnets, illus- trating the action of atomic forces t. Many modifications of these classical experiments have been suggested. In particular, a paper by R. Ramsey de- scribes interesting modifications of the original apparatus J. Actual apparatus illustrating the supposed structure of atoms can now be obtained ready for use from scientific instrument makers §. All these methods involve the repulsive forces between steel elements (needles or spheres) in a permanent magnetic field, together with the central attraction set up by a per- manent magnet. An important point contributing to the success of the experiment is that all the magnets, repre- senting electrons, have as nearly as possible equal pole strengths. Owing to magnetic reluctance and effects of demagnetization, these conditions are difficult to realize in practice without a considerable amount of care and ex- penditure of time. * Communicated by the Author. t J. J. Thomson, ' Corpuscular Theory of Matter * (1907), Chapter 6, pages 103 et seq. X It. R. Ramsey, "The Kinetic Theory of the Electron Atom." Pro- ceedings of the Indian Academy of Sciences, 1918. Phil. Mag - , vol. xxxiii. Feb. 1917, pp. 207-211. § W. M. Welch, Scientific Company, Chicago. 396 Prof. L. V. King on a Lecture-Room Section 2. The magnetic elements which form the essential feature of the apparatus to be described consist of a number of steel spheres or small soft-iron rods magnetized in a strong alternating field. One such model is shown diagrammatically in PL II. fig. 1, while fig. 2 shows the actual apparatus. The coil A consists of 340 turns of number 12 B. k S. copper wire (2 mm. diam.); inside radius of winding 8"8 cm., outside radius 13'5 cm., width of coil 3" 9 cm. Such a coil has a resistance of approximately 1*3 ohms and self-inductance of about 32 millihenries. It may be connected directly to a 110-volt 60-cycle A.O. circuit without overheating. In such cir- cumstances it draws a current of about 9 amperes. It is approximately of such dimensions as to give a maximum field strength at the centre of the coil. Placed over the opening of the coil is a large watch-glass B whose radius of curvature is approximately 25 cm. If available, an accurately ground concave glass mirror may be used to advantage. If, now, a supply of steel ball-bearings about 3 mm. in diameter is available, these may be placed on the concave surface B, where they will experience an attraction towards the lowest point approximately pro- portional to the distance. When the maximum current is passed through the magnetizing coil, the steel spheres will become A.C. magnetic doublets of very uniform magnetic moments. It will be noticed that the magnetic axis will always be very accurately along the direction of the mag- netic field, independently of the rolling motion of the balls. Furthermore, if the spheres are of fairly uniform quality and the field strength sufficiently great, the instantaneous mag- netic moments of these doublets will be equal in magnitude and phase. In these circumstances the steel spheres will repel each other with a force varying as the inverse fourth power of the distance, the constant of proportionality being accurately the same for all the spheres. With the attraction to the centre varying as the distance, it maybe expected that the magnetic elements will form remarkably symmetrical stable groupings. One such grouping is illustrated in PI. II. fig. 3 (a). It is obvious that by a very simple arrangement of lenses and mirrors this model atom may be projected on a screen. The concave surface B may, if desired, be mounted so as to allow of rotation, thus increasing the interest of the " atomic " arrangements. This experiment is extremely convenient for lecture-room purposes, as it requires no preparation and is Demonstration of Atomic Models. 391 always certain to give results which never fail to delight an audience. An interesting variant of this experiment is to make use of the arrangement of two coils described in Section 5 (figs. 5 & 6). A surface of clean mercury is placed midway between the two coils. A number of steel balls floating on this surface will repel each other as already described, and will all tend towards the centre, owing to the greater intensity of field. The remarkably regular arrangement taken up under these conditions is shown in fig. 3 (6). The damping is so slight that the system may be set into oscillation in various ways by means of external magnets, giving a good illustration of internal vibrations in the atom. It would, moreover, be possible with no very great expenditure of labour to deter- mine the frequency of various modes and compare the results with theoretical calculations. Section 3. The same apparatus may also be used to illustrate the motion of the molecules of gas or the Brownian movements. For this purpose an elongated piece of iron is employed, e. g. a short cylinder of iron or steel wire about 1 cm. in length by 1 mm. in diameter. In the alternating field of the coil such a magnet experiences a very strong torque, which vanishes when the axis lies along the direction ot the resultant A.O. field. If such a magnet is placed in a flat cylindrical glass vessel occupying the centre of the coil, ai.d the field suddenly applied, violent movements of the little iron rod will be observed. The instantaneous moments set up by the field will be sufficient to make the rod leave the surface on which it is resting and describe a trajectory under the combined effect of gravity and the magnetic field. At the termination of the flight, it will again strike the glass plate and will then receive an additional impulse made up of the magnetic torque and the elastic reaction at contact with the glass. This will start it on a new trajectory, and the process will be continued indefinitely until the rod makes contact with the plate at the termination of its flight in such a way that the instantaneous torque is zero. Then it stops dead with the axis pointing along the direction of the field. This is an event which happens very rarely. Several such rods enclosed within a glass vessel will keep in constant motion in a manner resembling the motion of molecules in a rarefied <:as. An interesting variant of this experiment is to insert .-hort steel wires along the diameters of small pith balls which hop around, describing flights in the glass vessel as if 398 Prof. L. V. King on a Lecture-Room they were animated with life. As before, the glass vessel and its contents may be projected on a screen, the resulting effect being illustrative of molecular movements. Section 4. — Experiments on Electrodynamic Repulsion. Owing to the distribution of the magnetic field around the coil employed in cbis experiment, the same apparatus is well suited to the demonstration of electrodynamic repulsion. For this purpose several plates of aluminium or copper should be cut with a radius approximately equal to the outer radius of the coil. Such a disk may be anchored by three strings fastened at equidistant points of the circumference so as to allow it to move vertically, with its centre over the axis of the coil, which is laid in a horizontal position. On applying A.C. circuit, the plate will float three or four centimetres above the coil. By placing a light iron rod (3 cm. x 1 mm.^ on the plate, the direction of the A.C. field is easily demonstrated, as shown in PI. II. fig. 4. It will be noticed that over an annular region bounded by the outer edge of the plate and a circle of half its radius, the lines of force are inclined at approximately 45° to the vertical. It is the reaction of the horizontal component of the A.C. field with the induced current due to the vertical component which causes the repulsion referred to. To demonstrate this, a circular plate may be cut up into several concentric rings and laid on a sheet of glass. When current is applied it is only the outer rings which are repelled, the force on the inner rings gradually becoming less, until that on the central disk in a practically uniform field perpendicular to its plane is practically nil. Iron filings poured on a glass plate laid horizontally over the coil assume an interesting laminar distribution, which again may be projected on a screen. The iron filings tend to arrange themselves in a series of vertical planes about 1 cm. high arranged radially. It is easily seen that this arrangement is due to the fact that under the influence of the alternating field, each of the radial planes represents a series of vertical A.C. magnets which repel each other. Their height is limited by the vertical stability of the plates under the combined effect of gravity and of the alternating field. Section 5. — Experimental Model of the Rutherford Atom. By using two coils of the dimensions already described, arranged with their planes horizontal at a distance apart equal to the mean radius (Helmholtz arrangement), it is Demonstration of Atomic Models. 399 possible to secure a fairly uniform field over a considerable area midway between the coils. Such arrangement (PI. II. figs. 5 & G) allows of interesting experiments on a model atom approximating more closely to modern ideas. A shallow circular basin of mercury is placed on an adjustable stand between the two coils. A number of steel pins with glass heads serve as the elements (electrons) for the model. It oue of these is placed with the glass head on the mercury surface, it will float in a vertical position and tend to move towards the centre ot the field, owing to the greater concen- tration of lines of force. This force towards the centre may be varied at will by adjusting the height of the mercury surface, or by placing rods of soft iron along the axis of the coils at adjustable distances above or below the mercury surface. If a second pin be floated on the mercury surface, it will repel the first with a force varying nearly as the inverse square law when the distance apart is not too great. A third pin may be added, when a triangular arrangment will be formed. Successive pins give the familiar series of regular polygons arranged in concentric rings. It is evident that the great advantage of the A.C. field is to make the mag- netical polarity of each of the pins very nearly equal, thus giving rise to a remarkable symmetry in the arrangements formed, as illustrated by figs. 7 (a) and 7(b) (PI. II.). As before, the experiment can be carried out in such a way that the various stable arrangements may be projected on a screen. It is extremely simple to demonstrate the apparatus at a moment's notice, the only precaution necessary being to use clean mercury so as to allow a great mobility of the floating pins on an uncontaminated surface. It is interesting to notice that rotation of the basin con- taining mercury does not disturb any particular stable arrangement, owing to the fact that the centrifugal force is accurately balanced by the change of slope of the para- boloidal mercury surface. The use of an A.C. field allows of the possibility of realizing positive electrons and a central nucleus, the law of forces between them being very nearly that of the inverse square and at the same time very exactly that corresponding to charges of ±e, ±2e, ±3^, etc. It is evident from fig. 5 (PI. II.), illustrating the model under consideration, that electrons may be represented by lengths of soft-iron wire of the same diameter arranged to move with both ends in the same plane at distances not too far apart compared with their length. In these circumstances we have repulsion according to the inverse square law, the charge — e being represented 400 Lecture-Room Demonstration of Atomic Models. by the average pole strength ±?n of each rod, which is extremely uniform. A nucleus of positive charge ne may be made up by taking 2n lengths of the same wire and inserting them in a small glass or aluminium tube, as shown in fig. 5, illustrating a nucleus of charge + 2e. In these circumstances, each of the rods representing electrons is attracted to the nucleus with a force varying nearly as the inverse square of the distance and proportional to nm x m } the average pole strength of each end of the rod being ztm. In order to realize this arrangement, the rods (about 7 cm. x 1 mm. diameter), representing negative electrons, should be suspended from silk fibres about 1 metre or more in length. By adjusting the position of the rods in the space between the coils, a position of neutral equilibrium may be found in which there is practically no tendency for the rods to move either towards the centre or radially outwards. Under the combined effect of gravity and of the magnetic field they seem to float in any position. When this adjustment has been made, the rods representing the nucleus should be set in position along the axis of the coils. The suspended rod representing the electron may then be projected so as to describe a path about the fixed nucleus, and a damped elliptic orbit will be observed, the nucleus being at one focus. . If two lengths of wire are used to make up a nucleus -\-e in the manner illustrated by fig. 5 (a), we obtain a model of the hydrogen atom which is dynamically stable. If we make up positive nucleus of charge 2e, represented by two pairs of iron rods, we obtain a model (fig. 5 (b)) of the ionized helium atom which is dynamically stable. If we introduce an additional iron rod representing an electron (fig. 5), and therefore a complete helium atom, it seems impossible to obtain a dynamically stable arrangement by any circumstances of projection. For instance, any attempt to reproduce the symmetrical oscillation suggested by Langmuir meets with failure, owing to the dynamical instability of this arrangement. It is obvious that further experiments along these lines, leading possibly to results of great interest, might be carried out by constructing large solenoidal coils to give a uniform A.C. field, in which circumstances the inverse square law of attraction and repulsion between electrons and nuclear charges ne (n=l, 2, 3, etc.) would be faithfully reproduced. L 401 ] XL. The Influence of the Size of Colloid Particles upon the Adsorption of Electrolytes, By Humphrey D. Murray, Exhibitioner of Christ Church, O.vford*. ^EVERAL workers have examined the influence of con- k^ centration upon the coagulation of colloidal solutions, hut references to the effect produced by alteration in the degree of dispersion are few and not very definite. Kruyt and Spek t examined the coagulation of colloidal arsenious sulphide, and found that the coagulative value of univalent ions increased with increasing dilution ; in the case of a divalent ion there was a slight decrease ; whilst for a ter- valent ion there was a rapid decrease in the coagulative value. Burton and Bishop + examined the coagulative values of various ions upon colloidal solutions of arsenious sulphide, copper, and gum mastic, and as the result of their experiments found that with univalent ions the concentration of the ion required for coagulation increased with decreasing concentration of the colloid, for divalent ions the concentra- tion of the ion was nearly constant, for trivalent ions the concentration of the ion varied almost directly with that of the colloid. More recently YVeiser and Nicholas § have extended these researches to colloidal solutions of hydrous chromic oxide, prussian blue, hydrous ferric oxide, and arsenious sulphide. They found in the case of the first three that the coagulative values of electrolytes tended to increase with dilution of the colloid, but the increase was less marked with electrolytes having univalent precipitating ions, and became more marked as the valency rose. Oden found that sols with ultramicroscopic particles are more sen-itive to electrolytes than those containing amicrons. The object of these experiments was to examine the influence of the size and uniformity of colloid particles upon the adsorption of electrolytes as measured by the minimal concentration for coagulation. For this it was necessary to obtain solutions of the same colloid prepared under identical conditions, but containing particles of different mean size. It was decided to employ Oden's method of fractional coagulation. The most suitable colloid to use, therefore, is one which, when first made, contains particles of markedly * Communicated by the Author. t Kruyt and Spek,*AV/. Zeit. xxv. p. 1 (1919). X Burton and Bishop, Jour. Phys. Chern. xxiv. p. 703 (1920). § Weiser and Nicholas, Jour. Phys. Chem. xxv. 742 (1921). Phil. Mag. S. 6. Vol. 44. No. 260. Aug. 1922. 2 D 402 Mr. H. D. Murray on Influence of Size of Colloid different size, and is stable when precipitated, redispersed, and dialysed. Gum mastic was found best to meet the requirements, and was used in the subsequent experiments. To show that the solutions employed were comparatively stable, the concentration of NaCl required to precipitate one of the fractions at the beginning and end of the experiments was measured and found to be : — Feb. 15th 433 millemols. Mar. 29th 439 Fractionation. One gram of finely-powdered picked gum mastic was dissolved in about 20 c.c. of alcohol, and poured slowly with vigorous stirring into one litre of distilled water. By this method seven litres of mastic solution were prepared. Oden recommends that in all cases NaCl should be used for the precipitation. With mastic this necessitates a very large concentration of salt, which appears to be strongly adsorbed, and comes slowly through the dialyser. It was thought better to employ HC1, which precipitates in smaller concen- tration. It was found convenient to separate the mastic into seven fractions with these concentrations of HC1 : — Concentration of HC1 Condition of Fraction, in millemols. Precipitate. I 0-1-1 trace II ri-l"4 good III 1-4-1-7 IV 17-2-0 V 20-2-3 VI 2-3-2-6 small VII. 26-2-9 trace The procedure was as follows: — 200 c.c. of the mastic N solution were mixed with a quantity of ^0^01 *° §^ Ye * ne required concentration, and then poured into the centrifuge vessels and allowed to stand for 60 minutes from the moment of mixing. It was then centrifuged at 3000 r.p.m. for 30 minutes. At the end of this time the supernatant liquid was poured off and the precipitate carefully shaken up with about 100 c.c. of distilled water. Fractions II. and VI. were retained until about 1500 c.c. of each had accumulated ; the other fractions were rejected. At the same time 1500 c.c. Particles upon the Adsorption of Electrolytes. 403 of mastic were completely coagulated with a concentration of 3*0 millemols. of HC1, and redispersed in an equal quantity of water. It appears below as solution B. It is to be expected that Fr. II. will contain particles of an average size greater than those in Fr. VI. and both will contain particles of more uniform size than those in solution B. The solutions after dispersion were kept in dialysers of parchment paper until the dialysate was uncontaminated with HC1. They were then placed in perfectly clean vessels of resistance glass fitted with a siphon, and a soda-lime tube attached to the air-inlet. The siphon pipes were closed by short pieces of rubber tubing and pinch cocks. Basis of Comparison. Any method of comparison between two or more solutions based upon the total masses of the disperse phase in unit volume is useless when applied to data due to adsorption. It is possible to take as a basis the number of particles in unit volume, or, what is probably more characteristic and capable of giving more directly comparable results, the total inter- facial surface in unit volume. The former may in most cases be ascertained by a direct count under the ultramicroscope. To evaluate the latter it is necessary, beyond this, to know the total mass of the disperse phase, which can be effected by weighing after evaporation to dryness, or by the methods of volumetric analysis. In addition it demands a knowledge of the density of the disperse phase, or of the specific gravity of the solution and of the dispersion medium. Perrin in his researches upon Brownian movement obtained the density of the mastic with which he was working by evaporating- a portion of his suspension to dryness and estimating the density of the solid mastic. This value (1'064) he found to agree admirably with the density as determined from specific gravity measurements. It seems uncertain, however, as Burton * has pointed out, whether it is justifiable to assume that the density of the particles in the ordinary colloidal solution of gum mastic is the same as that of the solid substances. Perrin, as a matter of fact, used a suspension of mastic which had been obtained by centrifnging the larger particles from a solution of mastic and rejecting the remainder. In the case of the present solution, it seemed desirable to determine the density of the particles directly, with a pvknometer. * Burton, ' Physical Properties of Colloidal Solutions,' 2nd Edition, p. 125. 1 D 2 404 Mr. H. D. Murray on Influence of Size of Colloid Concentration of Mastic. Thirty c.c. of the three solutions were evaporated slowly to dryness in a steam oven, and, as a mean of several deter- minations, gave the following weights of mastic in 10 c.c. of solution : — Weight found. Solution B -00463 gms. Fr. VI -00171 „ Fr. II -00339 „ Number of Particles. A true ultramicroscope was not used to count the particles, but a cardioid condenser, fitted to an ordinary microscope. The chief difficulty in work of this nature is to ascertain accurately the volume of the liquid within the field of view. A cell was made according to the recommendations of Siedentoff'*, the only alteration made being the substitution of heavy glass for fused quartz. Fluorescence due to the glass was not sufficient to render difficult the counting of the comparatively large mastic particles. The cell consists of a glass plate, 5 cm. in diameter and 1*0 mm. in thickness, provided with a circular groove. The portion enclosed by the groove, 1 cm. in diameter, was polished exactly 2 /jl deeper than the surface of the plate. This was used with a cover slip about *25 mm. in thickness. The cell was soaked in concentrated sulphuric and chromic acids, washed with water, and then passed through two solutions of re-distilled alcohol. It was finally flamed. The source of illumination was a Pointolight lamp, fitted with a condenser. All the solutions examined were diluted with water which had been carefully distilled and allowed to stand for a month undis- turbed. It contained on an average 1 particle in 20 counts in a volume of 14'1 x 10" 5 cu. mm. and could, there- fore, be considered optically pure to the degree of accuracy to which work was carried. All the solutions were contained in vessels of resistance glass, closed with corks covered with tinfoil. The method of procedure was to transfer, by means of a clean platinum loop, a very small drop of the solution to be examined to the central portion of the cell. The cover slip was laid on and pressed down until the Newton interference rings appeared at the edges. The dilutions were such that, when viewed with a convenient stop in the eyepiece, about three or four particles appeared * Siedentoff, Verhd. Deut. Phys. Ges. xii. p. 6 (1910). Particles upon the Adsorption of Electrolytes. 405 in the field of view. One hundred counts were taken at half- minute intervals, and the average number deduced from this. A few of the {(articles, especially in the case of Fr. II., tended to adhere to the walls of the cell, and to prevent any error due to this, the field of view was shifted five times during- each count. The results obtained were as follows : — Solution. Dilution. Soln.B.. x72G Fr. II. . . x 396 Fr.VL.J X396 I Ob- ject iA-e. '4 mm. f.l. Eye- piece. jDiam. of , Field of vieAV. X 12 -30 mm. X 12 -30 „ Xl8 I -20 „ Volume of Field ^ '% of view. t-, ,'. , Particles, 14'1 X 10 - 5 mm. 14-lxlO- 5 „ 6-28 X 10 - 5 „ 4-1 30 4-0 Density of the Particles. The density of the solutions was determined with an accurate pyknometer in a thermostat at 17*2° C. The weighings agreed to *0002 gm. Fr. VI. was too dilute to give accurate results. The specific gravity of each solution rose slightly during dialysis, owing probably to the removal of adsorbed or dissolved alcohol. This rise continued for about five days. The weighings were made at the end of ten days. The dialysis was then continued in more efficient dialysers made by Soxhlet thimbles impregnated with collodion, but the specific gravity remained constant. As a mean of four weighings for each solution, the following values for the density of the particles were obtained : — Soln. 3. = 1-195. Fr. II. =1-186. As a mean the density of the mastic was taken to be 1*190. This value is considerably higher than the density of the mastic in bulk, owing possibly to changes occurring either on dispersion, or coagulation. Perrin * states that he observed the density of his carefully washed granules apparently to rise in salt solutions, and this may account in part for the difference. Size of Particles. From these three sets of data — the number of particles in unit volume, the total mass of mastic in unit volume, and the density of the particles— it is possible to calculate the mean * Perrin, Ann. C'him. Phys. xviii. p. 5 (1909). 406 Mr. H. D. Murray on Influence of Size of Colloid radius r of the particles in each liquid. 01' these three measurements that of the density seemed possibly least accurate, but, as it occurs in each calculation, the relative sizes remain unchanged. Solution. Soln.B. Fr. II.. Fr. VI. No. of Particles in mm. 3 (=»). 211x10 s 84-2 xlO 5 252 x10 s 1 Total Massi of Particles in 10 c.c. •0046 gm. •0034 ,. •0017 „ 1 Mean volume of one particle. 1-83X10- 2 3-40x10-2 •57x10-2 J" 3 Radius (=r). r 2 X»XlO -3 . •164 p •201 p ■111 JU 564 340 307 r 2 xn is a measure of the interfacial surface in unit volume. Borjeson * has successfully combined the principle of gilding metal particles with observation of the rate of sedi- mentation of the particles so gilded, to measure the size of the original particles. He failed to obtain successful results with gelatine and gum arabic sols. An attempt to apply this method to the mastic solutions as a check on the results obtained also met with failure. It appears therefore unsuit- able for organic colloids. Rate of Coagulation. It has been customary to fix an arbitrary time during which the colloid solution is allowed to stand after the addition of the electrolyte and before the amount of coagulation is measured. Burton f allowed the solutions which he examined to stand ten hours ; and again J the more dilute solutions which he examined were left for "some days." Weiser and Nicholas § allowed the solutions under examina- tion to stand for twenty-four hours. In some preliminary experiments the writer found that abnormal results were obtained with a dilute solution owing to the fact that the time elapsing before examination was too short to permit of coagulation with the minimal quantity of electrolyte. This led to an examination of the actual rate of coagulation. Two solutions were employed, one being ten times more dilute than the other. The results w ere as follows : — * Borjeson, Koll. Zeit. xxvii. p. 18 (1920). t Burton and Maclimes, Jour. Phys. Chem. xxv. p. 517 (1921). X Burton & Bishop, Jour. Phys. Chem. xxiv. p. 703 (1920). § Weiser and Nicholas, Jour. Phys. Chem. xxv. p. 742 (1921). Particles upon the Adsorption of Electrolytes, 407 2 S m o 5 'o - CR9 S! w2 c ° - 00 tT — 2- 9 aq £L <s E9 SI jT O CI -*• CO 5 ft US S 0* M 75 b o' g o" 3 x O t-f (-"O g '-'Ccai.aj-ai-^cco Or005H05rf.OO 5 ee 5 p p; H? S S S 3 « 2 S aT 2 B o g CD R --.».. ^ j _ S-' CTC ct> u_i Q £ ~ o s O _ i— ■ 2 § 3.dS" Ci O 3 rf r3- s o s *a s - Ei^ Br" c p " S-" " p'g. 3 «> -s S: «8 ^ •Sgs CO JL P5QTQ CD O P O 4 ^ m o ® P <-s SI * to O £-' !£ o tr cr z cr» s - ^ s ^_ o p » 3 J2 1 ^ DO PS O o *r CO si £ rf^ O J=3 s Cr 1 - ; : ; j' » P P. o C cfq B) ? g o" o O CD O CD P hj 02 CO c 408 Mr. H. D. Murray on Influence of Size of Colloid It is obvious that the rate of coagulation decreases con- siderably with decreasing concentration of the mastic. The point of complete coagulation was taken to be " clear with large flocks." These flocks do not necessarily settle to the bottom of the vessel ; some adhere to the side. The interior of the liquid, however, appears quite clear. It is noticeable that the flocks adhering to the sides are more numerous upon, if not confined to, the side of the vessel away from the source of daylight illumination. It is apparent that, in order to arrive at the point of complete coagulation in the case of the more dilute solution, it is necessary to leave it undisturbed for a good many days, a course of action to which there are several objections, apart from that of: mere convenience, in carrying out a series of numerous determinations. During this time external influences, such as chemical action at the surface of the particle, have more time to show themselves. These objections can be obviated by centrifuging the solution after a definite time at a constant speed. The rate of coagulation is made up of two factors, the rate of aggregation of the particles and that of settling of the aggre- gates so formed. By centrifuging, the influence of the latter is reduced to a minimum, and we arrive at a truer measure of the former. An examination of this method shows that it is possible to obtain complete coagulation after a reasonable length of time. The solutions were treated in the way to be described, and the following tables (and figs. 1 and 2) show the minimal concentrations of A1 2 (S0 4 ) 3 and NaCl at various intervals after the moment of addition for complete coagulation. Similar results were obtained with BaCl 2 . It will be seen that the minimal concentration of electro- l} r te decreases rapidly with time until from 12 to 22 hours after mixing, thereafter, it remains fairly constant. In order, however, to ensure reaching the true end-point, the solutions in the subsequent experiments were allowed to stand 48 hours after mixing and before centrifuging. Several workers have pointed out that the rate and method of addition of the electrolyte affect the end-point. Weiser and Middleton * devised an apparatus, by the use of which they obtained concordant results, and a modification of it made by the writer was found to give equally good results. In order to ensure perfect cleanliness, it was made of glass throughout. The modified apparatus consisted of * Weiser and Middleton, Jour. Phys. Chem. xxiv. p. 30 (1920). Particles upon the Adsorption of Electrolytes. 409 1 o o Limina ncentra of NaC g o' 3 B O s 4 J-; P 5' C ' o ^ O | l_l — o c o I y o Q V ■" t* 4 ,— ' » Cn Ot o 8 8 B 3' - cT » a X o^ JO V I— 1 Cn OS 33 O CO 3* c o — 1 B ~ 3" — ! B n o to cC J^ l O Cn ° ° LO B 3 " l c GO CO rf>. •— Or c o Oi r S CD 2 SI c OD -5 CO ro c: c o tO — 4- E3 3 - CD X o_ CO -J OS to o» o o j^. 30 — - 2 B £_ P o O o p d 3C - 0Q s 3 2; P o o P »»§ ^2 t- 1 ,r-2 g- 3 OD^f -• 3* *°&£ cj"' o • 3 2 * — v ' o k 12 i - ET. an? o 3 )_i i-i o o o ^ p V V tO fcO o o 3l 3 3 f -1 3 s cd 3 3 C^ c CO V V to to Oi o o © B ;=: 3 ~ CO V o B 00 V to O HJ © Di ? — B 3 ST 5 CO - 3 c 09 to CO CO CO _i 5 3* pi 3 | CD CO H^ CO en co to — 3t J2 3- - aT o £ CD h^ ci Ot 00 *» 5 DO 3 - 2 o L p P a> o | -^ '— > O P crq p c-f- o (-/J w o 60 i-rs FJ P GO 1— i HH o * ^2 o 410 Mr. H. D. Murray on Influence of Size of Colloid two vessels, one slightly smaller than the other, and fitting by a ground-in joint inside the larger. The smaller vessel has a slightly higher inner cylindrical vessel, the base of which is concentric with that of the outer vessel and fused to it. The electrolyte solution is placed in the inner vessel and the colloidal solution in the annular space, both having been previously rinsed out with their respective solutions. The larger vessel is placed over the smaller, and the whole inverted and left for 30 seconds '•J* n e i £ °aoo 1 v 1 Vv v^..^ , 10% ■4 — ) g\g i .,100% ,., i Time in hours- after mixing W to drain. By this means a sudden and complete mixing of the two solutions is obtained. The mixed solution is then poured into a vessel of hard glass and corked. The whole apparatus, as were all the vessels with which the mastic solution came in contact, is made of hard glass and was steamed out between each series. The experiments were conducted with 10 c.c. of the mastic solution at the required dilution. Into the inner vessel was poured enough water to make the volume of the elec- trolyte solution up to 5 c.c, and then the latter solution was added at a convenient concentration. To determine Particles upon the Adsorption of Electrolytes. 411 one end point for a given concentration of mastic and a given electrolyte, four solutions were made up with a fairly wide difference of concentration in each solution, so as to give a large bracket. After standing 48 hours and centrifuging for half an hoar at 2000 r.p.m., at which speed there was no sedimentation of the pure mastic solution, four more solutions were mode up, in which the concentrations of electrolyte were such as to cover the interval between the 3-2 H Fig I \ \ \ \ L \ V \ \ \ v \ ,.100% -xJ2&- 9 : \l lit 36 1*8 Time in hours after mixing two concentrations in the first determination, within the limits of which the end point was observed to lie. The process was repeated until the limit of observation was reached, and eventually gave two concentrations of which it was possible to say that one definitely caused complete coagulation, and one did not. The end point was taken as the mean of these tw^o concentrations. The observation of the solutions was made by daylight against a black back- ground. The size of the final bracket of concentration varied directly with the concentration of electrolyte necessary for coagulation, and the results were therefore more accurate with trivalent ions than with monovalent. The results were as follows (Series 5, 6 and 7, and figs. 3, 4 and 5). 412 Mr. H. D. Murray on Influence of Size of Colloid CV. ft »o : : t^ ^ ' © d ft o o . b- os • CO rn© • CO rH 6 OOiO ft O tp Tf o CM rH i-H CM £ ft . o »o l^ o • " cq o o a O iO th © © Tf) CO CM CM CM CO <M co < ■+3 d d bf) & >Oh -f C Tf CM CM CM c, o O C5 | ft b-Tf b- CO CM CM CM l> CO w W « © H ft OS J>. OS C/J b- cb CM CM CM © d ■4J W ^ S3 ^ O o r - «- GO rHrH o 13 ,1 £ -u <D e8,3 O S 2 O O r~ OS o „ O^ Eh .1^ i-h d & iO ^ • CO OSOS • CO rH d A 00 CO Tf r~ CO 00 O © i— 1 CM d ft XHN © X) 00 OS 0* Tf o PQ -+■= d rB » w> hp ' — ' CO b- Tf as CO OS t- OS o CO 1 lO h-H t> SQ H P3 ft 1Q OS CO rH b- OS t^- OS cyj © co d m 03 ~o co Eh Et< r- ' v '* O CJ_ 03 "IS o 03 g s u o s o-~ rH 1i q 5 ^ "acq 13 CJ ft oo . o © • b- £~ ' co rH d ©© © ft © co as o © © b- CM d ft © ©© rfi oo os »r iQ ir; CO CO „ d ft ©© © O CO CO O iO © a CO * -r= c 03 ,_j ft © © © t-^CO ^f< qc; ir~ HH Tf IO GJ co o tH 1 O 1 ! i—3 d ! t— i ft . .© t> CO • -OS • -TJH co W ifj Ph 1 SQ d 1 ft © © © CO © IO Tf Tt< Tf CO CO I _d co n. B. ... II VI 03 W. Cm hh | O "tr — ' i JH o O c » -u O ^ : ■J3 o ! ■*3 a a £ c O o a , i Q.rH 03 Liminal NaC d p 00 CO © CO CM CM CO d ft iO iO -— i CO t- Tf Tf 00 •^ d CO 00 rH Mi rH © © © rH CM d ft — 1 rH CM iC OS CO 1—1 CO CM d ft CO CO CM 00 rH © CO i— 1 rH rH CO CO <D O CS «W o C3 © © CO ft CM CO CM 02 © Tf CM rH rH c3 O I C3 o «-l-l ^H ft CO © CO CV © IO Tf 1 ->-=> t~ CM rH rH d © 1 HH Tf d 1 ft rH-HTf © 00 © CO CO —* — 1 co iO d ft © © Tf b- CM © : b- CO CM CM © © d i -*-: a S w H rH ! o 6^> . r- O CO Ph tM C3 n © o O X O P X P=l w Particles upon the Adsorption of Electrolytes. 413 0^* fj£ ^a ^a Cod^u Unt -Al M).- i IS Q J 3o 150 3oo 350 \ v £ K F 3 c 2 "- 2 5 Fiol - 1 KM -r ■MAK? - IVaCl s . 30 >o I M 50 1 30 414 Size of Colloid Particles and Adsorption of Electrolytes. Discussion of Residis. It will be seen that, under the conditions imposed and within the limits of the experiments, a comparison of the data obtained, upon the basis of the total interfacial surface in unit volume, leads to uniformity in the curves. Such uniformity is not to be observed when the comparison is based upon the mass of the disperse phase, or the number of particles, in unit volume. It appears that adsorption is very largely conditioned by the amount of interfacial surface exposed. It is to be noticed, however, that the minimal concentration of electrolyte is higher throughout for the fraction containing small particles than for that containing large particles. This may be brought about in two ways. The smaller particles may bear a higher charge per unit area of their surface, or the critical value to which their charge must be reduced before coagulation begins may be lower than in the case of the larger particles. The latter explanation is more probably correct, since it is known that the surface tension of large particles is greater than that of small ones. It seems probable, if the existence of a critical potential difference for coagulation between a particle and the dispersion medium be admitted, that this should be lower in the case of small particles which have less tendency to adhere, and should thus permit of a greater freedom of approach between the particles. If the former explanation were correct, we should expect a separation of the particles according to size in an electric field ; but this is contrary to experience, the particles move at the same rate independently of their size. According to the Helmholtz theory of the electrical double la}^er, this effect is due to equal density of *the charge upon unit area of the surface. It appears probable, therefore, that the smaller particles have a lower critical potential difference for coagulation. The behaviour of the solution containing mixed particles of different size is in some respects curious. With both A1 2 (S0 4 ).3 and NaCl the curve representing the coagulation of this solution is more flattened relatively than the other two curves. A lack of uniformity in ths size of the particles appears to render the solution less sensitive to change in concentration, in the case of coagulation by univalent and trivalent ions. Notices respecting New Books. 415 Summary* (a) A separation of the particles present in a suspension of gum mastic has been effected by OdeVs method of fractional coagulation. (b) The density of the particles, and the mass of mastic and the number of particles in unit volume have been measured, and from them the interfacial surface in unit volume calculated. (<•) The variation of the minimal concentrations of A1 2 (S0 4 ) 3 , BaCl 2 , and NaCl to coagulate solutions containing particles of different mean size with change in concentration of the solutions has been investigated. (d) It has been shown that uniformity in comparison of the results can be obtained upon the basis of the interfacial surface in unit volume. It has also been shown that, upon this basis of comparison, small particles require a higher minimal concentration of electrolyte than large particles. In conclusion I should like to thank Dr. A. S. Russell for his valuable advice and assistance, and Mr. H. M. Carleton for kindly putting at my disposal the microscopical apparatus required. Christ Church Laboratory, Oxford, May 1 5th, 1922. XLI. Notices respecting New Books. Basic Slags and Rock Phosphates. By Gr. Scott Robertson. Pp. xiv + 112, 8 plates. 1922. Cambridge Agric. Monographs. Cambridge University Press. 14s. net. r pHE value of scientific investigation of the results accruing from -*- the use of phosphatic dressings on crop-production is obvious to all, but it gains in emphasis when, as Sir E. J. Russell points out in a preface to the above book, agriculturists have to realize that the composition of basic slag has undergone much change in consequence of the enforced modifications in the processes of steel manufacture. We would go farther than Sir E. J. Russell and 416 Notices respecting New Books. say that even if the war had not given an impetus to the change over from the basic Bessemer and acid open-hearth processes, economic considerations would none the less have demanded the development of the basic open-hearth production of steel from low-grade iron-ores. " This result" (to quote from the preface) " is, of course, distinctly awkward for the agriculturist who sees a valuable fertilizer disappearing, and being replaced by one which is more costly and at first sight seems to be nothing like as good." After a review of the various scattered experiments on the use of rock phosphates and basic slags hitherto undertaken, Dr. Scott Robertson describes in detail the Essex experiments carried out in the winters of 1915, 1916, 1918, and 1919 under the auspices of the East Anglian Institute of Agriculture. The soils treated were those of the Chalk, London Clay, and Boulder Clay, and varied considerably in mechanical and chemical composition. The yields of hay and clover were correlated with the rainfall, and it was found that the drier the season, the greater was the increase in production due to the use of phosphates. The botanical results are also given, the crowding-out of the weeds and the covering of bare areas with grass being noteworthy. Dr. Robert- son's main conclusion is that for root crops and late harvests with high rainfall, rock phosphates will prove a suitable substitute for the high-grade Bessemer basic slags. The careful records and correlations were made personally by Dr. Robertson at consider- able inconvenience and discomfort, and under most difficult circumstances. They are therefore the more valuable, and do him the greater credit. The latter part of the book is concerned with investigations of the large yields resulting from the use of basic phosphates. From botanical analyses it is evident that the open-hearth fluor-spar slags of low solubility are less effective than the non-fluor-bearing and therefore more highly soluble slags. The effects of the temperature and texture of the soil on the accumulation of nitrates, on the soil bacteria, and on the acidity and lime-require- ment are clearly expounded, and the deductions emphasized by means of abundant statistics. Altogether, the work constitutes a most valuable contribution to agricultural knowledge. It is a pity that the publishers cannot retail this book of 112 pages and 8 plates for less than 14s. P. G. H. B. Phil. Mag. Ser. 6, Vol. 44, PI. II, Fig. 7(a). Fig. 7 (b). Phil. Mag. Ser. 6. Vol. 44, PI. II. Fig. 7 («). Fig. 5(«). a -L n n T Fig. 7 (6). THE LONDON, EDINBURGH, and DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [SIXTH SERIES] SEPTEMBER 1922. XLIL The Disintegration of Elements by a. Particles. By Sir E. Rutherford. F.R.S., Cavendish Professor oj Experimental Physics, and J. Chadwick, Ph.D., Clerk Maxwell Scholar, University of Cambridge *. IN a former paper f we have shown that long-range particles, which can be detected by their scintillations on a zinc-sulphide screen, are liberated from the elements boron, nitrogen, fluorine, sodium, aluminium, and phos- phorus under the bombardment of a rays. The range of these particles in air was greater than that of free hydrogen nuclei set in motion by u particles. Using radium G as a source of a rays, the range of the particles varied from 40 cm. for nitrogen to 90 cm. for aluminium, while the range of free hydrogen nuclei under similar conditions was about 29 cm. Previous experiments J by one of us had indicated that the long-range particles from nitrogen were deflected in a magnetic field to the extent to be expected if they were swift hydrogen nuclei ejected from the nitrogen nucleus by the impinging a particle. The nature of the particles from the other five elements was not tested, but it seemed very probable that the particles were in all cases H nuclei which were released at different speeds depending on the nature of: * Communicated by the Authors. t Rutherford and Chad wick, Phil. Mag. vol. xlii. p. 809 (1921). X Rutherford, Bakerian Lecture, Proc. Roy. Soc. A, vol. xcvii. p. 374 (1920). P/u7.J/a^.Ser.6.Vol.44.No.261.&?p*.1922. 2E 418 Sir E. Rutherford and Dr. J. Chadwick on the the element and on the velocity of the incident a particle. Under the conditions of the experiment, these H nuclei could only arise from a disruption of the atomic nucleus by the action of the a particles. Attention was also drawn to the remarkable fact that in the case of the one element examined, viz. aluminium, the particles were liberated in all directions relative to the incident a particles. In the present paper we shall give an account of ex- periments to throw further light on these points and to test whether any evidence of artificial disintegration can be observed in the case of other light elements. Magnetic Deflexion of the Particles, In the course of this work, the microscope used for the counting of scintillations has been further improved. For the present experiments it was essential, in order to obtain a sufficient number of scintillations per minute, that the area of zinc-sulphide screen under observation should be greatly increased without diminution of the light-gathering power of the microscope system. Following the suggestion of Dr. Hartridge, a modified form of Kellner eyepiece was constructed. A planoconvex lens of about 7 cm. focal length was placed so as to render the rays of light from the objective approximately parallel, and the image so formed was viewed through an eyepiece consisting of a similar lens and an eye-lens of 4 cm. focal length. Used in conjunction with the old objective, Watson's Holoscopic of 16 mm. focal length and *45 numerical aperture, this system gave a field of view of a little more than 6 mm. diameter. A rectangular diaphragm was placed in the eyepiece ? limiting the field of view to an area 6 mm. x 4'9 mm. Our previous system had a field of view of 83 sq. mm. area, so that the new microscope, under similar conditions, gave about three times the number of scintillations of the old. The precautions adopted in counting were similar to those described in our previous paper. The method of measuring the magnetic deflexion of the long particles was very similar to that described by one * of us in the Bakerian Lecture of 1920. The experimental arrangement is shown in fig. 1. The source of a. rays was placed at R and was inclined at an angle of 20° to the horizontal. The lower edge was level with the face of a brass plate S which acted as a slit. * Rutherford, he. cit. Disintegration of Elements by u Particles. 419 The distance from the centre of the source to the farther edge of the slit was 2*95 cm. The carrier of the source and slit was placed in a rectangular brass box between the poles of an electromagnet, the field being perpendicular to the length of the slit. A current of dry oxygen was circulated through the box during the experiment. Fig. 1. If 4- vS 1 3^ An extension piece L, projecting 1*7 cm. beyond the edge of the slit, was fixed to the carrier in order to increase the amount of deflexion of the particles issuing from the slit. In the end of the box was a hole 1 cm. wide and 2 cm. long covered with a sheet of mica of 3*62 cm. stopping-power. The ZnS screen was fixed on the face of the box, leaving a slot of 1 mm. depth in which absorbing screens could be inserted. The source H was a brass disk of 1*2 cm. diameter coated with the active deposit of radium. Its initial 7-ray activity was usually equivalent to about 40 mgm. Ra. The materia], the particles from which were to be investi- gated, was laid directly on the source if in the form of foil, or if in the form of powder dusted over its face. The experiment consists in obtaining an estimate of the deflexion of the particles falling on the screen by observing the effect of a magnetic field on the number of scintillations near the line E, the edge of the undeflected beam of particles. The position of the microscope was fixed in the following- way : — After placing the source of a rays in position, hydrogen was passed through the box. The u rays could then strike the ZnS screen, and the edge E of the beam was clearlv defined. The microscope was adjusted so that the 2 E 2 420 Sir E. Rutherford and Dr. J. Chadwick on the edge o£ the beam of scintillations appeared a little above a horizontal cross-wire in the eyepiece of the microscope, marking the centre of the field of view. When the magnetic field was applied in such a way as to bend the a particles upwards (called the positive direction of the field), the edge of the beam is deflected downwards in the field of the microscope and the scintillations appear only in the lower half. When the field was applied in the opposite direction (negative field), the edge of the beam moved upwards in the field of view. The strength of the magnetic fields used in the experiments was always such that the whole field of view was covered with scintillations when the negative magnetic field was applied. In the experiments on the magnetic deflexion of the long-range particles, the number of particles is far too small to give a band of scintil- lations with a definite edge. It is clear, however, that if the particles are positively charged, the number of scintil- lations observed with the negative magnetic field will be greater than the number observed with the positive field, and that the ratio of these numbers will give a measure of the amount of deflexion of the particles. By determining this ratio for the long-range particles and comparing it with that for projected H particles of known velocity, we can obtain an approximate value for the magnetic deflexion of the long-range particles. The general method of the reduction of the observations is perhaps best shown by an account of the experiments on the particles from aluminium. Experiments on Particles from Aluminium. After fixing the position of the microscope in the way described above, an aluminium foil of 3*37 cm. stopping- power was placed over the source. Dry oxygen was passed through the box, and a mica sheet of 10 cm. stopping-power was inserted in front of the ZnS screen. The total absorption between the source and screen was then equivalent to 30 cm. of air. The scintillations observed were consequently due to long-range particles from the bombarded aluminium ; the ranges of the particles under observation varied from 30 cm, to 90 cm._, the average range being about 45 cm. Counts of the numbers of scintillations observed with positive and negative fields due to an exciting current of 6 amps, were then made. The mean ratio of the numbers with a — field to those with a + field obtained from several experiments was 3*7. The observations were repeated with Disintegration of Elements by a Particles. 421 a field due to an exciting current of 4 amps. ; the corre- sponding ratio was 2*1. When the source had decayed to a small fraction of its initial value, the aluminium foil over the source was removed and a thin sheet of paraffin wax put in its place. The mica sheet in front of the ZnS screen was replaced by a sheet of 3'4 cm. stopping-power, making the total absorption equi- valent to 16 cm. of air. The scintillations observed on the screen were now due to H particles ejected from the paraffin wax of ranges between 16 cm. and 29 cm., the average range being about 22 cm. The ratio of the numbers of scintillations for — and + fields was determined for an exciting current of 4 amps, and found to be 3*2. It appears from these results that the long-range particles from aluminium of average range 45 cm. were less deflected by the same magnetic field than H particles of average range 22 cm. ; and that in the magnetic field due to a current of 6 aiups.^ which was 1*34 times the intensity of the field due to 4 amps., they were more deflected than were the H particles in the latter field. To a first approximation we may say that the value of mvje for particles from aluminium of range 45 cm. is 1'23 times greater than that for H particles of range 22 cm. This result is clearly consistent with the view that the particles from aluminium are H nuclei moving with high velocity; for, assuming that the range of the H particle is pro- portional to the cube of its velocity, the velocity of a particle of range 45 cm. is 1*27 times that of a particle of 22 cm. range. These experiments show, therefore, that the particles from aluminium carry a positive charge and are deflected in a magnetic field to the degree to be anticipated if they are hydrogen nuclei moving with a velocity estimated from their range. While there can be little doubt that the particles are hydrogen nuclei, it is very difficult to prove this point definitely without an actual determination of the velocity and value of e/m of the particles. Our knowledge of the relation between the range and velocity of complex charged particles is too indefinite for purposes of calculation. On the other hand, if we assume, as seems a priori probable, that the ejected particle is the free nucleus of an atom, it is possible to show with some confidence that only a particle of mass 1 and charge 1 can fit the experimental results. Additional evidence as to the value of mvje of the particles from aluminium was obtained by comparing their magnetic deflexion with that of the a particles of S'Q cm. range emitted by thorium C. In this experiment the source R was a very 422 Sir E. Rutherford and Dr. J. Chadwick on the weak source of the thorium active deposit obtained by ex- posing a disk to thorium emanation. Hydrogen was passed through the box, and sufficient absorbing screens were inserted in front of the ZnS screen to cut out the 5 cm. a particles of thorium G. The numbers of a particles falling on the screen for — and + fields due to an exciting current of 6 amps, were counted, and the ratio of these numbers was found to be 2'4. Comparing this ratio with those found for the long-range particles, we see that the value of mv/e for the latter is about 0'8 of that for a particles of 8'6 cm. range, i. e. about 3*4 x 10 5 e.m. units. The calculated value, assuming that the particles are H nuclei and that their velocity is pro- portional to the cube root of the range, is about 3*7 X 10 5 ,e.m. units. Considering the difficulty of the experiments, the agreement is satisfactory. Experiments on Phosphorus and Fluorine. Measurements similar to the above have also been made on phosphorus and fluorine. In the case of phosphorus, a thin layer of red phosphorus was dusted over the face of the source. The total absorption in the path of the particles was about 35 cm. ; the range of the particles under observation varied therefore from 35 cm. to the maximum range of 65 cm., the average being about 45 cm. The ratio of the numbers of scintillations for — and + fields due to current of 4 amps, was 2*0. In the case of fluorine finely powdered calcium fluoride was dusted over the source. Previous experiments have shown that no long-range particles are emitted from calcium. The total absorption in the path of the particles was about 30 cm. The maximum range of the particles from fluorine is approxi- mately 65 cm., and the average range of the particles falling on the screen was around 40 cm. The ratio of the numbers of particles observed for — and ■+■ fields due to an exciting current of 4 amperes was 2*5. It is clear from these results that, within the error of experiment, the particles liberated from phosphorus and from fluorine are bent in a magnetic field to approximately the same extent as the particles from aluminium. We may conclude, therefore, that these particles also are H nuclei moving with high speed. We have not examined the particles from boron and sodium in this way, but there seems no reason to doubt that they also consist of H nuclei. Disintegration of Elements by a Particles. 423 The Ranges of the H Particles. In the experiments described in our previous paper only two elements, nitrogen and aluminium, were investigated in any detail. The other elements were examined in a qualitative manner, but it was shown that the ranges of the liberated H particles were in every case greater than 40 cm. of air. The ranges of the particles from these elements — viz. boron, fluorine, sodium, and phosphorus — have now been determined more accurately. Attention has been drawn to the remarkable fact that the H particles liberated from aluminium appeared not only in the direction of the incident ot particles but also in the reverse direction. The number of particles emitted in the backward direction was of the same order of magnitude as for the forward, but the maximum range in the backward direction was smaller, being 67 cm. as against the 90 cm. range of the forward particles, for a particles of 7 cm. range. Some experiments with nitrogen showed that the number of H particles emitted in the backward direction was very small at absorptions of more than 18 cm. of air. We have repeated these experiments and extended them to include the other elements boron, fluorine, sodium, and phosphorus, with the result that we find that in every case the H particles emitted on disintegration of the nucleus escape in all directions, the maximum range in the backward beino- less than in the forward direction. Fig. 2. A A BLJ<t0 !l!ljllll|lt!l|l||i|lll The experimental arrangement for the measurement of the ranges of the forward particles was the same as that described in our previous paper. The apparatus used in the investigation of the particles in the reverse direction differed from this in the arrangement of the source, and is shown in the diagram (fig. 2). The source of a particles was carried on a rod passing with a sliding fit through a stopper which fitted tightly into the 424 Sir E. Rutherford and Dr. J. Chad wick on the brass tube T of 3 cm. diameter. The end of this tube was provided with a hole 7*5 mm. in diameter, closed by a silver foil of 3*75 cm. air equivalent. The zinc sulphide screen S was fixed on the face of the vessel leaving a slot in which absorbing screens could be inserted. The apparatus was placed between the poles of an electromagnet to reduce the luminosity produced in the screen by the /3 rays. The source R was a silver foil of 4'15 cm. stopping-power coated on one side only with the active deposit of radium. Its initial 7-ray activity was in most experiments equiva- lent to about 30 mg. Ra. The inactive side of the silver foil faced towards the ZnS screen. The distance of the source from the screen was generally about 3*5 cm., but could be varied, and its position read off on a scale. The elements to be examined could in most cases only be obtained in the combined state. The powdered compound was heated in vacuo, and a film prepared by dusting on to a foil smeared with alcohol. The screen thus prepared was placed immedi- ately behind the source. As in our previous experiments, a stream of dry oxygen was circulated through the apparatus. In all cases, except that of nitrogen, the maximum range of the particles emitted in the backward direction was greater than the range of free hydrogen particles, so that no complication arises from the presence of hydrogen in the silver foils or other materials in the path of the a particles. In the case of nitrogen, however, as our previous experi- ments had shown, the range of the backward particles is much less than that of free hydrogen particles, and it was consequently necessary to allow for the " natural" effect, i. e. for the H particles arising from hydrogen contamination of the source and screens in the path of the a rays. It was found inconvenient to use gaseous nitrogen for these experi- ments, and a suitable screen was prepared by sifting a thin layer of powdered paracyanogen, CyST x , on to a gold foil. The scintillations observed on the ZnS screen when the film of paracyanogen was placed against the source were due to the " natural " particles from the source and screens, together with those which came from the nitrogen in the paracyanogen. On taking away the film of C Z N X the natural particles alone were counted. In some experiments a film of paraffin wax was placed against the source. The natural effect remained the same, showing that even if the film of paracyanogen contained a large amount of hydrogen the number of free H particles scattered to the ZnS screen by the walls of the vessel was negligible. Figure 3 shows the type of results obtained in these Disintegration of Elements by « Particles. 425 experiments. The ordinates represent the number o£ scintil- lations observed per minute per milligram of activity of radium C, measured by 7 rays ; the abscissae, the stopping power for a. rays of the absorbing screens, expressed in terms of centimetres of air. The dotted curve A gives the natural effect observed when the screen of paracyanogen was absent or replaced by a film of paraffin wax ; the full curve B the effect when present. The difference of these curves there- fore represents the effect due to the nitrogen in the para- cyanogen. It will be seen that the maximum range of the Fig-. 3. 20 1-5 0-5 \ Forward and Ba from Afitt cAward Particles ■oyen. \ ~-JV \B \cC ■ -»» % :0 25 30 A bsorption m cms of sir 40 backward particles from nitrogen is about 18 cm. Curve C is the absorption curve for the particles emitted by nitrogen in the forward direction. In the following table are given the maximum ranges of the particles liberated from the elements which show the disintegration effect, for both forward and backward directions. Element. Forward rang* Boron 58 Nitrogen 40 Fluorine 65 Sodium 58 Aluminium 90 Phosphorus ••• 65 Backward range. cm. 38 18 48 36 67 49 426 Sir E. Rutherford and Dr. J. Chadwick on the It should be pointed out that the ranges of the forward particles from boron, fluorine, sodium, and phosphorus may be subject to considerable error, owing to the use of a film of powder as the bombarded material. The particles of maximum range are produced on the surface of the grains of powder, and therefore to find the true range the size and air equivalent of the grains of powder must be known. For the ranges given above it has been assumed that the grains were uniform in size and an average value of the air equivalent of the film of powder has been calculated from its weight per sq. cm. The ranges so determined are obviously somewhat less than the true ranges. The ranges of the backward particles are, of course, not subject to this source of error. It was observed that the number of particles liberated from the different elements appeared all to be of the same order of magnitude when allowance is made for the differ- ence in range. In our original experiments we found that the number ojj particles from boron was somewhat smaller than the numbers from the other elements, but this was due to the use of an irregular film. Using a film of more finely powdered boron it was found that the number of particles from boron was about the same as from the other elements. Examination of other Elements. In our former experiments we examined all the light elements, with the exception of the rare gases, as far as calcium. Of these only the six elements of the above table were found to emit H particles in detectable amount under the bombardment of & rays. As was pointed out in that paper, the atomic masses of these elements can be represented by 4n + a where n is a whole number, a result which receives a simple explanation on the assumption that the nuclei of these elements are composed of helium nuclei of mass 4 and hydrogen nuclei. On the other hand, some of the light elements which gave no detectable number of H particles also had atomic masses given by 4w + a. It was thus a point of great importance to repeat the examination of these elements with the improved microscope, and to search, if possible, for the emission of particles of shorter range than free H nuclei. In some cases it was only possible, on account of hydrogen contamination of the materials, to observe at absorptions greater than 30 cm. of air, while in others the observations were carried well within this range. Lithium was examined as oxide and as metal, a thin sheet of the latter being obtained by pressing molten lithium Disintegration of Elements by a Particles. 427 between two steel plates in an atmosphere of carbon dioxide. No evidence was found of any particles of range greater than 30 cm. Owing to hydrogen contamination of the Li and Li 2 the observations at smaller ranges were not decisive. Observations in the backward direction revealed no detectable number of particles of range greater than 14 cm. Beryllium was examined as the powdered oxide, and there was again no evidence of the emission of particles of longer range than 30 cm. in the forward direction or 15 cm. in the backward. Magnesium was examined with a sheet of the metal and also with a screen of powdered magnesium. There was no evidence of long-range particles. For silicon a screen of powdered silicon and a thin sheet of quartz were used. With the sheet of quartz it was possible to make observations in the forward direction at absorptions as low as 17 cm. The scintillations observed were due entirely to the natural H particles. To CO, Chlorine had been previously examined in the form of various chlorides. These observations were repeated, and the results confirmed the conclusion that particles of greater range than 30 cm. were not liberated in any detectable amount. In order to pursue the observations within the range of free H particles a special series of experiments was 428 Sir E. Rutherford and Dr. J. Chadwick on the carried out. A glass apparatus, similar in design to the standard apparatus, was used. The details will be clear from the diagram (fig. 4). In order to avoid the bombardment of the glass walls and consequent liberation of H particles the inside of the tube was lined with platinum foil. The surfaces of the brass plate B and of the rod carrying the source were protected from the action of the chlorine by a coating of hard pitch. The stopcocks and grouod-joint were lubricated with a brominated grease. The source of a rays was a platinum foil coated with radium active deposit. Pure dry chlorine was prepared by heating gold chloride, AuCl 3 , contained in the tube A, and was passed over P 2 5 before entering the vessel T. As an additional precaution a little P 2 0$ was placed in the vessel itself. When the source was placed in position the air was removed by pumping and washing with dry carbon dioxide. Carbon dioxide was then let in to atmospheric pressure and the natural H particles were counted at absorptions varying from 16 cm. to 30 cm. The carbon dioxide was then replaced by chlorine, and the scintillations at similar absorp- tions were observed. The chlorine was then allowed to be reabsorbed by the gold chloride and carbon dioxide let in again. In this way counts on the chlorine were included between counts of the natural particles, and any traces of adventitious hydrogen could be allowed for. The results showed no evidence of the liberation of H particles from chlorine in the range examined, i. e. at absorptions more than 16 cm. of air. Discussion of Results. For convenience of discussion the atomic numbers and the masses of the isotopes of the elements from hydrogen to potassium are given in the following table. Of these elements aluminium is the only one which has not yet been examined for isotopes, but it appears likely that it is a pure element of atomic mass 27. With the exception of helium, neon, and argon, all the elements in the table have been tested to see whether H nuclei are ejected by the action, of a particles. The six active elements, as they may be termed for convenience, are underlined. Disintegration of Elements by a Particles. Table T. 429 Element. £ t01 » ic a amber. h r~ He 2 Li 3 Be 4 B 5 c" 6 N 7 O 8 Fl 9 Ne 10 Atomic Masses. 1-008 400 6,7 9 10,11 12 14 16 19 20,22 Element. ,£ toD ? io .Number. 2ii IT - Mg 12 Al 13 Si 14 P 15 S 16 CI 17 A 18 K 19 Atomic Masses. 23 24, 25. 26 27 28.29 31 32 35,37 36,40 39,41 An examination of the table shows that the active elements may be classified in different ways : — (1) Active elements are odd-numbered elements in a regular sequence of numbers, viz., 5, 7, 9, 11, 13, 15. (2) The atomic masses of the active elements are given by 472 + a where n is a whole number ; a = 3 for all the elements except nitrogen, for which it is 2. (3) With the exception of boron, which has two isotopes (10, 11), the active elements are all pure elements. We have seen that no evidence has been obtained that the preceding element lithium (3), and the succeeding elements, chlorine (17) and potassium (19), show any trace of activity under a-ray bombardment, although they are odd-numbered elements and the masses of their isotopes are given by 4n-f- a. Magnesium and silicon, which are even-numbered, but which contain isotopes of mass 4?i -t- 1 or 4n + 2, show no sign of activity. There thus appears to be no obvious general relation which differentiates active from inactive elements. The activity starts sharply with boron and ends abruptly with phosphorus. It is a very unexpected observation that neither lithium nor chlorine shows any certain evidence of activity in the emission of either long-range or short-range particles. It is of interest to consider whether any deduction can be made as to the structure of these nuclei in the light of these experimental facts. In our previous paper it was pointed out that the H nuclei 430 Sir E. Rutherford and Dr. J. Chadwick on the liberated from the active elements probably existed as satellites circulating in orbits round the main nucleus. In the case of an effective collision of an a particle with such a nucleus, part of the momentum of the a particle is com- municated to the central nucleus, but the satellite is sufficiently distant from the latter to acquire enough momentum and energy to escape from the system. It was shown that such a point of view offers a general explana- tion of the variation of the velocity of the expelled H nuclei with the speed of the a particle and also of the escape of the H nuclei in all directions. The chance of ejecting an H satellite at high speed from a nucleus is much smaller (for nitrogen, for example, about 1/20) than the chance of setting a free H nucleus in correspondingly rapid motion. It appears therefore that the release of the satellite only takes place under certain restricted conditions of the collision of the v particle with the nucleus. If the H satellites were present in lithium and chlorine and were very lightly bound to the nucleus, it is to be anticipated that the number released by the a rays would be of the same order of magnitude as if the H nuclei were free. As this is found not to be the case, we may conclude that neither lithium nor chlorine has any lightly bound satellites in its nuclear structure. The complete absence of long-range particles from these elements shows that the H satellites, if they are present at all, are strongly bound to the main nucleus. If, for example, the satellite revolves very close to the nucleus, the a particle may only be able to give such a small part of its momentum to the satellite that it is unable to release it from the system. It does not, however, seem likely that the forces binding a satellite would vary greatly in passing from phosphorus to chlorine. It seems more probable that the general structure of the chlorine nucleus differs in some marked way from that of the group of active elements. The H nuclei may perhaps be definitely incor- porated into the main nuclear system, so that the a particle has no chance of concentrating its energy upon a single unit of the nuclear structure. In a similar way it seems probable that lithium must differ widely in structure from the suc- ceeding element boron. The facts brought to light in these experiments indicate that the nuclei even of light elements are very complex systems and illustrate how difficult it will be to find any simple and general rule to account for the variation in structure of successive elements. It has been pointed out that, with the exception of the first Disintegration of Elements by a. Particles. 431 element boron, all the active elements are " pure " elements, i. e., have no isotopes. This may be of some significance in differentiating between the structure of active and inactive elements. The absence of isotopes indicates that, as regards mass, there is only a narrow range of stability of the nucleus for a given nuclear charge ; the addition or subtraction of an equal number of H nuclei and electrons leads presumably to an instability of the nuclear system. In the case of lithium and of chlorine, which form isotopes, the forces binding the nuclei together may consequently be very different from those in the case of the pure active elements. If there is any significance in this point of view, it would indicate that H satellites are only present in pure odd- numbered elements ; but, as we have seen, boron is an exception to this rule. In comparing the phenomena shown by the six active elements, it seems at once clear that nitrogen occupies an exceptional position in the group. Not only is the range of the expelled H nuclei the smallest of all the group, but the ratio of the ranges in the two directions is markedly different from those shown by the other elements. It is natural to connect this anomalous behaviour with the fact that the mass of the nitrogen nucleus is given by 4w + 2, while the rest of the group are of the class 4n + 3. The slower speed of ejection of the particles from nitrogen at first sig;ht suggests that the H satellite is more lightly bound than in the case of the other elements. This suggestion is, however, not borne out by calculation of the distribution of momentum among the three bodies involved in the collision, viz., the a particle, the H satellite, and the residual nucleus. Jn our previous paper, we showed that the distribution of momentum could be calculated on certain assumptions from the observations of the ranges of the expelled nuclei in the forward and reverse directions of the a particle. It was supposed that the law of conservation of momentum holds, and that the sum of the energies of the H particle and the residual nucleus was the same whether the H particle was liberated in the forward or backward direction. It follows from these assumptions that the relative velocity of the H nucleus and the residual nucleus is the same in the two cases. The results of this calculation for the group of active elements are collected in the following table (Table II.). -54 Y 323 Y 2-52 Y 42% 1-44 V 454 Y •78 V -13 % --10Y 3-89 Y 2-00 Y 35 % 1-42 Y 5-16 Y •56 Y 6% •78 Y 523 Y •88 Y 42% 432 The Disintegration of Elements by ol Particles. Table II. Distribution of Momentum. H particle. Residual Nucleus. a Gain in me " Forward. .Backward. Forward. Backward, particle. Energy- Boron 202 Y - 1'75 Y Nitrogen 1*78 V -1'32 Y Fluorine 2-10 Y - T89 Y Sodium 202 Y -172 Y Aluminium... 234 V -211 Y Phosphorus... 210 Y -1-89 V VUT 5-13 Y 76 Y 15% The momenta are expressed in terms of the initial velocity V of the a particle. The initial momentum of the a particle, and consequently the sum of the momenta of the three bodies after collision, is therefore 4V. Momenta in the direction of the incident a particle are taken as positive, momenta in ths opposite direction as negative. The percentage energy, gained from the nucleus as a result of the disintegration, is given in the last column, in terms of the initial energy of the ol particle. It will be seen that in the case of nitrogen a considerable part of the momentum of the a particle is communicated to the main nucleus, a much greater part than in the cases of the adjacent elements boron and fluorine. This indicates that the H satellite of nitrogen is in relatively close proximity to the main nucleus. It will also be noted that while for the other elements there is a gain of energy from the disruption varying from 6 per cent, for sodium to 42 per cent, for boron and aluminium, for nitrogen there is a loss of energy of 13 per cent. It is apparent from the above table that the distribution of momentum among the three bodies varies considerably for the different elements, but, in the absence of any definite evidence of the validity of the theory on which the calcula- tions are based, it seems inadvisable to discuss these differences in any detail at the present stage. Cavendish Laboratory, June 20, 1922. L 433 ] XLIII. The Distribution of Electrons around the Nucleus in the Sod/ ion and Chlomne Atoms. By W. Lawren'CE Bragg, M.A., T.lx.S.. Langworthy Professor of Physics, The University of Manchester ; R. W. James, M.A., Senior Lecturer in Physics, The University of Manchester ; and C. H. Bosanquet, M. A., Balliol College, Oxford*. 1. TN two recent papers f in the Philosophical Magazine i the authors have published the results of measure- ments made on the intensity of: reflexion of X-rays by rock-salt. The mathematical formula for the intensity of reflexion, as calculated by Darwin J, involves as one of its factors the amount of radiant energy scattered in various directions by a single atom when X-rays of given amplitude fall upon it. The other factors in the formula can bo evaluated. By measuring the intensity of reflexion experi- mentally we can therefore obtain an absolute measurement of the amplitude of the wave, scattered by a single atom, in terms of the amplitude of the incident radiation. This measurement is of considerable interest, because it may throw some light on the distribution of the electrons around the nucleus of the atom. We regard the wave scat- tered by the atom, as a whole, as the resultant of a number of waves, each scattered independently by the electrons in the atom. A formula first evaluated by J. J. Thomson is used in order to calculate the amplitude of the wave scattered by a single electron. If an incident beam of plane polarized X-rays consists of waves of amplitude A, then the amplitude A' at a distance R from the electron in a plane containing the direction of the incident radiation, and at right angles to the electric displacement, is given by A R mc 2 {) Here e and m are the charge anc [ m ass of the electron in electromagnetic units, and c is the velocity of light. What we measure experimentally is the resultant ampli- tude of the wave-train scattered in various directions by a number Z of electrons in the atom. If all the electrons were * Communicated by the Authors. t Phil. Mag. vol. xli. March 1921 ; vol. xlii. July 1921. X C. G. Darwin, Phil. Mag. vol. xxvii. pp. 315-675 (Feb. and April 1914). Phil. Mag. S. 6. Vol. 44. No. 261. Sept. 1922. 2 F 434 Prof. W. L. Bragg and Messrs. James and Bosanquet : concentrated in a region whose dimensions were small com- pared with the wave-length of the rays, then the resultant Z e 2 amplitude would be equal to ^ — ^ smce the scattered _tx Tnc wavelets would be in phase with each other in all directions. It is found experimentally that the measured amplitude tends to a value which is in agreement with the formula at small angles of scattering, but that at greater angles it falls to a very much smaller value. This is to be accounted for by the action of interference between the waves scattered by the electrons in an atom, which are distributed throughout a region whose dimensions are large compared with the X-ray wave-length. It is an easy matter to calculate the average amplitude scattered in any direction by a given distribution of electrons around the nucleus. Here we are attempting to solve the reverse of this problem. The experimental results tell the amplitude of the wave scattered by the sodium and chlorine atoms through angles between 10° and 60°. We wish to use these results in order to get some idea of the manner in which the electrons are distributed. 2. In addition to Darwin's original mathematical treat- ment, the question of the effect on X-ray reflexion of the distribution of electrons around the atom has been dealt with by W. H. Bragg *, A. H. Compton f , and P. Debye and P. Scherrerif. W. H. Bragg considered the interpretation of the diminu- tion in the intensities of reflexion by a crystal as the glancing angle is increased, due allowance being made for the arrangement of the atoms. He concluded that "an ample explanation of the rapid diminution of intensities •is to be found in the highly probable hypothesis that the scattering power of the atom is not localized at one central point in each, but is distributed through the volume of the atom." He did not regard the experimental data then available as sufficient to justify making an estimate of the distribution of the electrons. These data indicated that the intensity of reflexion fell off roughly as . 2 ~ (0 being the glancing angle), and he showed that a density of distribution of the electrons could be postulated which * W. H. Bragg, Phil. Trans. Roy. Soc. Series A, vol. ccxv. pp. 253-274, July 1915. t A. H. Compton, Phys. Rev. vol. ix. no. 1, Jan. 1917. t P. Debye and P. Scherrer, Phys, Zeit. pp. 474-483, July 1918. Distribution of Electrons in JSa and CI Atoms. 4IJ5 accounted for this law, just as an illustration of the appli- cation of the principle involved in considering spatial distribution. A. H. Compton used the experimental results obtained by W. H. Bragg in order to calculate the electron distribution. W. H. Bragg showed that the intensity of reflexion is a function of the angle of reflexion alone, when allowance has been made for the arrangement of the atoms in the crystal, and he determined the relative intensity of reflexion by a number of planes in rock-salt and calcite. Compton cal- culated from these values the relative amplitudes of the waves scattered by the atoms in different directions, by means of the reflexion formula of Darwin, and proceeded to test various arrangements of electrons in order to find one which gave a scattering curve agreeing with that found experimentally. He supposed that the electrons were rotating in rings, governed by Bohr quantum relationships In sodium, for example, he placed four electrons on an inner ring, six on the next ring, and a single valency electron on an outer ring. In chlorine the rings contained four, six, and seven electrons respectively. Compton found that these atomic models gave a fair agreement with W. H. Bragg's results. Debye and Scherrer came to the same conclusion as to the significance of intensities as regards electron distribution which was implied in W. H. Bragg's work and stated more fully by Compton. They considered two interesting cases. The first was that of the lithium fluoride crystal. They compared the intensity of reflexion by planes where the fluorine and lithium atoms reflected waves in phase with each other, with that by planes where these atoms acted in opposition to each other. The relative amplitudes at any TT -i- T i angle for such planes may be expressed by the ratio ^ — ^> where F and Li are the amplitudes contributed by the fluorine and lithium atoms respectively. Their figures indicated Ui r • that the limiting values of ^ — p at zero angle of scattering is 1*5, signifying that a valency electron has passed from the lithium to the fluorine atom I ^ =1*5 ). Their intensities of reflexion were measured by the darkening of a photographic plate in the powder method of analysis which these authors initiated. In view of the 2F2 436 Prof. W. L. Bragg and Messrs. James and Bosanquet : difficulties o£ estimating intensities in this way, of the few points which they obtained on the curve for the ™ x • ratio, of the difficulties in interpreting intensities which we have discussed in our papers, and of the large extrapolation which they had to make in order to get the limiting value of -j= — j-. , we feel that their results cannot be regarded as r — .Li proving that the transference of the valency electron has taken place. The fact of the transference is supported by much indirect evidence, and their conclusion is probably correct. Debye and Scherrer also compared the intensities reflected by various planes of the diamond, and concluded that the electrons in the carbon o atoms were contained within a sphere of diameter 043 A, assuming a uniform distribution throughout this sphere. In all the above cases, the results were obtained by com- paring the relative intensities of reflexion by various faces. The results which we have obtained, and which will be used to calculate the distribution of electrons in sodium and chlorine, are, on the other hand, absolute determinations. The intensity of reflexion was compared in each case with the strength of the primary beam of X-rays, so that the absolute efficiency of the atom as a scattering agent could be deduced. In a paper on "The Reflection Coefficient of Monochro- matic X Rays from Bock Salt and Calcite " *, Compton made comparisons of the incident and reflected beam, for the first order reflexion from cleavage faces of these crystals. He obtained results for rock-salt which were rather less than those which we afterwards obtained for a ground face, but he noted that the effect was increased by grinding the face. In our notation the results were Compton ^ - -00044 + '00002 ;XaCl(100). B.J. and B. ^ = v 00055 As Compton surmised, and as we have found experimentally, this figure for the efficiency of reflexion has to be modified considerably to allow for the extinction factor. The difference * A. H. Compton, Phys. Rev. vol. x. p. 95, July 1917. Distribution of Electrons in Na and CI Atoms. 437 between his results and ours is accounted for by the extinction or increased absorption of the rays at the reflecting angle. Compton pointed out that the reflexion factor was of the order to be expected from Darwin's formula, but did not use the value he obtained to solve the electron-distribution problem. 3. For the sake of convenience of reference, the formula which forms the basis of all the calculations is quoted below. Let the intensity I of a beam of homogeneous X-rays, at a given point, be defined as the total energy of radiation falling per second on an area of one square centimetre at right angles to the direction of the beam. If a crystal element of volume dV, supposed to be so small that absorption of the rays by the crystal is inappreciable, be placed so that it is bathed by the X-rays, and if it is turned with angular velocity co through the angle at which some plane in it reflects the X-rays about an axis parallel to that plane, the theoretical expression for the total quantity of energy of radiation E reflected states that ■^ = NV F «.4- 4 i±^.-B-n-»iV . (2) I sin 2d m 2 c 4 2 v ' = QdV. In this expression N= Number of diffracting units per unit volume*. A, = Wave-length of X-rays. 6 = Glancing angle at which reflexion takes place. e = Electronic charge. m = Electronic mass. c = Velocity of light. The factor e~ Bsm " (the Debye factor) represents the effect of the thermal agitation of the atoms in reducing the intensity of reflexion. The factor F depends on the number and arrangement of the electrons in the diffracting unit. At = it would have a maximum value equal to the total number of electrons in the unit, and it falls off owing to interference as 6 increases. The experimental observations have as their object the determination of Q in absolute units. In practice we cannot use a single perfect crystal so small that absorption is * Xo account is taken here of the " structure factor." The diffracting units are supposed to be spherically symmetrical as regards their diffraction effects. 438 Prof. W. L. Bragg and Messrs. James and Bosanquet r • inappreciable. We use a large crystal consisting of num- bers of such homogeneous units and deduce, from its reflecting power, the reflecting power Q per unit volume of the units of which it is composed. The assumptions made in doing this are by no means free from objections, and will be discussed later in this paper. Taking this to be justifiable, however, our experimental results yield the value of Q for rock-salt over a wide range of angles, and from them the values of F C i and F Na follow directly. These values are shown in fig. 1. Fur. 1. 18 1 \ \ \ % 12 8 \ \ 'I \ 4 *P) '(c) 0-3 S/o A (a) F C1 corrected for Debye factor, (c) F Na corrected for Debye factor. (b) F cl uncorrected „ „ (d) F Na uncorrected „ ,, 4. We must now consider more closely the significance of the factor F. The most simple case is that of a crystal con- taining atoms of one kind only. Parallel to any face of the Distribution of Electrons in Na and CI Atoms. 439 crystal we can suppose the atoms all to lie in a series of planes, successive planes being separated by a distance d. We get the nth order spectrum formed at a glancing angle by the reflexion from such a set of planes if 2d sin = n\. This spectrum represents the radiation diffracted by the atoms in a direction making an angle 26 with the incident beam, and it is formed because in this particular direction the radiation scattered by any pair of atoms lying in suc- cessive planes differs in phase by 2mr. Thus the amplitude of the beam scattered in this direction is the sum of the amplitudes scattered by all the neighbouring atoms taking part in the reflexion. Let us consider the contribution to the reflected beam of a group of atoms lying in a reflecting plane. To obtain the amplitudes of the reflected wave, we sum up the amplitudes contributed by the electrons in all the atoms, taking due account of the fact that the electrons do not in general lie exactly in the reflecting plane and so contribute waves which are not in phase with the resultant reflected wave. By symmetry, the phase of the resultant wave will be the same as that reflected by electrons lying exactly in the geometrical plane passing through the mean positions of all these atomic centres. The phase of the wave scattered in a direction 6 by an electron at a distance x from the plane differs from that of the resultant wave by an amount — -.2?sin 6. A, We will suppose that there is in every atom an electron which is at a distance a from the centre, and that all direc- tions of the radius joining the electron to the atomic centre are equally likely to occur in the crystal. In finding the effect of these electrons for all atoms (M in number) of the group, we may take it as equivalent to that of M electrons distributed equally over a sphere of radius a. It can easily be shown that, if x is the distance of an electron from the plane, all values of x between +a and —a are equally likely for both cases. Such a shell scatters a wave which is less than that scattered by M electrons in the plane in the ratio sin 6 , — — -, where * 4tt (/)= --a sin 0. 440 Prof. W. L. Bragg and Messrs. James and Bosanquet : The average contribution of the electron in each atom to the F factor is therefore —7— , and not unity as it would be if the electron were at the centre of the atom. If there are n electrons at a distance a from the centre of the atom, their contribution to the F factor would be sin 6 , Q . n -f < 3 > Any arrangement of n electrons at a distance a from the centre of the atom, provided that all orientations of the arrangement were equally probable, would make the same contribution to the F factor. Foe example, eight electrons arranged in a ring about the nucleus would give the same value for F as eight electrons arranged at the corners of a cube, or eight electrons rotating in orbits lying on a sphere of radius a. This illustrates the limitations of our analysis, which cannot distinguish between these cases. We can only expect to get information from our experimental results as to the average distance of the electrons from the atomic centre, and this for the average atom. Suppose now that any atom contains a electrons at a distance ?\ from the nucleus, b at a distance r 2 , c at a dis- tance r B . . . n at a distance r n , then the value of F for the average atom would be given by F =a !i^ + 6 si ^ + o%i 3 + ... + n ^i". . (4) 9i 92 93 9« Thus, given the distribution of the electrons on a series of shells or rings, we can calculate the value of F for any value of 6. The problem we have to solve here, however, is the converse of this. We have measured the value of F for a series of values of #, and wish to determine from the results the distribution of the electrons. We have seen above that there is no unique solution of this problem, but we can get some idea of the type of distribution which will fit the experimental curves. In order to do this, we suppose the electrons to lie on a series of shells, of definite radii r 1; r 2 , .... and determine the number of electrons a, b, c on the various shells which will give values of F corresponding to those observed experimentally. Suppose, for example, we take six shells uniformly spaced over a distance somewhat greater than Distribution of Electrons in Na and CI Atoms. 441 the atomic radius is expected to be. For any given value of 6 we have </>l </>2 </>3 </>t $5 +/^ (5) We chose from the experimental curve six values of 6 evenly spaced over the range of values at our disposal, and for each of these values read from the curve the value of F. Since definite radii have been assumed for the shells, the values of — j-^~ , etc., can be calculated for each value of 6. 9i Hence, for each value of 0, we have an equation involving numerical coefficients and the quantities a, b, c, d, e,f\ so that if six such equations are formed we may calculate these quantities. If Z is the total number of electrons in the atom w r e have Z=a+b + c + d + e+f, ...... (6) and this will be taken as one of our equations (corresponding to # — 0). In calculating the results for sodium and chlorine we have assumed the atom to be ionized, and have taken Zci = 18 and Z^ a = 10. It will be evident that this method of solution is somewhat arbitrary, and that the results we get will depend on the particular radii assumed for the shells. By assuming various radii for the shells, however, and solving the simultaneous equations for the number of electrons on each, we find that the solutions agree in the number of electrons assigned to various regions of the atom. As a test of the method of analysis, a model atom was taken which w T as supposed to have electrons arranged as follows : — 2 on a shell 0"05 A radius. 5 „ „ 0-35 3 „ „ 0-70 The F curve for this model was calculated. Then the simul- taneous equations for the electron distribution were solved, just as if this curve had been one found experimentally. This was done for two arbitrarily chosen sets of radii, taken out to well beyond the shell at 0*70 A. 442 Prof. W. L. Bragg and Messrs. James and Bosanquet : The comparison between the two analyses (dotted curves) and the atom model we started with (continuous curve) is shown in fig. 2. The abscissas represent the radii of the shells in A, the ordinates the total number of electrons inside a shell of that radius. When the limits of the atomic structure are reached, the curve becomes horizontal at the value 10, corresponding to the ten electrons. The analyses not only indicate with considerable accuracy the way in n3 fl 4 Fig. 2. xr._ 1-5 sphere, measured 2-0 Angstrom units. 2-3 which the electron-content grows as we pass to spheres of larger radii, but also tell definitely the outer boundary of the atomic structure. Both give a number of electrons very nearly equal to zero in the shells outside O70 A. 5. The F curves for sodium and chlorine can be solved in the same manner. We have expressed our results in two ways. First, we have supposed the electrons to be grouped on shells. The numbers of electrons on each shell, and the radii of the shells, have been so adjusted as to give the best possible fit to the experimental curves. In the case of sodium it is found that a fit can be obtained with two shells, and in the case of chlorine with three shells. The Distribution of Electrons in Na and CI Atoms. 443 numbers of electrons on each shell, and the radii of the shells, are as follows : — Sodium, 7 electrons on a shell of radius 0'29 A. o ,, ,, j, U'/o ,, Chlorine. 10 electrons on a shell of radius 0*25 A. 5 ., . ., „ 0*86 „. 3 ., „ „ 1-46 ., Secondly, we have solved the simultaneous equations for the distribution in shells with several sets of radii, and drawn a smooth curve through the points so obtained in such a way as to represent the density of distribution of the electrons as a continuous function of the distance from Fig. 3. 1 \ I 1 \ / \ \ 02 0-4 0-6 0-8 1-0 Distance from centre of atom in Angstrorr the atomic centre. The density P is so defined thatgPdr is the number of electrons whose distance from the centre lies between r and r+dr. The curves which we obtain for sodium and for chlorine are shown in figs. 3 and 4. The 444 Prof. W. L. Bragg and Messrs, James and Bosanquet : total number of electrons in the atom is represented by the area included between the curves and the axis. Fig. 4. i . Distance from centre of atom in Angstrom units. The following table shows the agreement between the F curves found experimentally and those calculated from the electron distributions : — Table I. — Sodium. Sin 6. 01. 0-2. 0-3. 0-4. 0-5. ^Observed 8-32 540 3-37 2-02 076 pj Shells (0;29A| g . 56 5-59 333 219 0-93 1 ^ Smooth Curve ... 8*S7 5-40 3-20 1-91 1-00 Table II. — Chlorine. Sin0. 0-1. 0-2. 0- 0-4. 05. Observed 12*72 7-85 5-79 4-40 3-16 773 5-90 4-61 2-69 7-80 5-55 410 3-20 f 0-25 A \ F { Shells \ 086 V 12-53 | I T46 j ^Smooth Curve ... 12-70 6. We have also made an approximate calculation ©f the F curve to be expected from an atom of the type pro- posed by Bohr *, In the ionized sodium atom containing * Nature, cvii. p. 104 (1921). Distribution of Electrons in Na and CI Atoms. 445 10 electrons, two are supposed to describe circular one- quantum orbits about the nucleus, while, of the remaining* eight, four describe two-quantum circular orbits and four two-quantum elliptical orbits. We have calculated the size of these orbits from the quantum relation ship and the charges ; this can only be done very approximately, owing to the impossibility of allowing for the interaction of the electrons. We take the following numbers : — Radius of 1 quantum ring 0*05 A. 2 „ 0-34 „ Semi-major axis of ellipses * ... 0*42 ,, To get a rough idea of the diffracting power of such an atom, we suppose, first, that the orientation of the orbits is random so that the average atom has a spherical symmetry, and also that the periods of the electrons in their orbits are so large compared with the period of the X-rays that we need not consider the effect of their movements. The calculation of the effect of the circular orbits offers no difficulties. To allow for the effect of ihe ellipses, the- following method was used. The elliptical orbit was divided into four segments, through each of which the electron would travel in equal times. It was then assumed that, on the average, one of the four electrons describing ellipses would be in the middle of one of these segments. This gives four different values of the radius vector, corresponding in the average atom to four spherical shells of these radii. AYe thus calculate the value of F for an atom having 2 electrons on a shell of radius 0*05 A. IT. 4 .. „ „ 0-34 ., 1 „ „ „ 0-27 „ 1 .. „ „ 0-55 „ 1 „ „ „ 0-70 „ 1 n .<, » 0*78 ,, * The elliptical two-quantum orbit of a single electron about the sodium nucleus would have a semi-major axis equal to the radius of the two-quantum circle. We have used the larger value 0-42 to make some allowance for the fact that part of the orbit lies outside the inner electrons, so that the effective nuclear charge is reduced. 446 Prof. W. L. Bragg and Messrs. James and Bosanquet : This gives the following figure for F^ a : — Sin 9. 0-1. 0-2. 0-3. 0-4. 0'5. F calculated F observed . . •73 5-04 376 .2-53 1-80 •32 5-40 3-37 2-02 0-76 The agreement, of course, is not perfect, but one must remember that no attempt has been made to adjust the size ■of the orbits to fit the curve. The method of calculation too is very rough, although it must give results of the right order. The point to be noticed is that the curve is quite of the right type, and there is no doubt that an average distri- bution of electrons of the nature given by such an atom model could be made to fit the observed value of F quite satisfactorily. 7. The points which appear to us to be most doubtful in the above analysis of our results are the following : — (a) We have assumed that each electron scatters inde- pendently, and that the amount of scattered radiation is that calculated for a free electron in space according to the classical electromagnetic theory. It is known that for very short waves this cannot be so, since the absorption of 7 rays by matter is much smaller than scattering would account for, if it took place according to this law. On the other hand, the evidence points towards the truth of the classical formula in the region of wave-lengths we have used (0-615 A). (b) We have used certain formulae (given in our previous papers, to which reference has been made) in order to calculate the quantity we have called Q in equation (2) from the observed intensity of reflexion of a large crystal. Darwin * has recently discussed the validity of these formulae. The difficulty lies entirely in the allowance which has to be made for "extinction" in the crystal. X-rays passing through at the angle for reflexion suffer an increased absorption owing to loss of energy by reflexion. Darwin has shown that this extinction is of two kinds, which he has called primary and secondary. If the crys- talline mass is camposed of a nuaiber of nearly- parallel homogeneous crystals, each so small that absorption in it is inappreciable even at the reflecting angle, then secondary extinction alone takes place. At the reflecting angle the * Phil. Mag. vol! xliii. p. 800, May 1922. Distribution of Electrons in Na and CI Atoms. 447 X-rays suffer an increased absorption, because a certain fraction of the particles are so set as to reflect them and divert their energy. We made allowance for this type of extinction in our work, and Darwin concludes that our method of allowance, while not rigorously accurate mathe- matically, was sufficiently so for practical purposes. Primary extinction arises in another way. The homo- geneous crystals may be so large that, when set at the reflecting angle, extinction in each crystal element shelters the lower layers of that element from the X-rays. Darwin has calculated that this will take place to an appreciable extent for the (100) reflexion if the homogeneous element is more than a few thousand planes in depth. A large homogeneous element such as this does not produce an effect proportional to its volume, since its lower layers are ineffective, and a crystal composed of such elements would give too weak a reflexion. Our method of allowing for extinction will not obviate this effect. We cannot be sure, therefore, that we have obtained a true measure of Q for the strong reflexions. The F curve may be too low at small angles. It is just here that its form is of the highest importance in making deductions as to atomic structure. Until this important question of the size of the homogeneous elements has been settled, we must regard our results as provisional. (c) The allowance for the thermal agitation of the atom (the Debye factor) is only approximate ; it depends on a few measurements made by W. H. Bragg in 1914. In order to see how much error is caused by our lack of knowledge of the Debye factor, we have calculated the electron distribution without making any allowance for it. The result may appear at first rather surprising ; the electron distribution so calcu- lated is almost indistinguishable from that which we found before, when allowance for the Debye factor had been made. This is so, although the factor is very appreciable for the higher orders of spectra, reducing them at ordinary tem- peratures to less than half the theoretical value at absolute zero. The difference which the factor makes can best be shown by comparing the radii of the shells which give the best fit with (1) the F curve deduced directly from the expe- rimental results, (2) the F curve to which the Debye factor has been applied. 448 Prof. W. L. Bragg and Messrs. James and Bosanquet : (1). (2). Radius (without allowance for thermal agitation). Radius (with allowance for thermal agitation). odium. — Seven electrons . 0-31 0-29 Three electrons . . . 0-79 0-76 hlorine. — Ten electrons . . . 0-28 0-25 Seven electrons... 0-81 0-86 Three electrons.. 1-46 1-46 A little consideration shows the reason for this. The form of the F curve at large angles is almost entirely decided by the arrangement of the electrons near the centre of the atom. A slight expansion of the grouping in this region causes a large falling off in the intensity of reflexion. This is shown in the analysis by the slight increase (O02 to O03 A) in the radius of the shell which gives the best fit to the uncorrected curve. '1 he effect of the thermal agitation is to make the electron distribution appear more widely diffused ; however, the average displacement of the atom from the reflecting plane owing to its thermal movements is only two or three hundredths of an Angstrom unit at ordinary temperatures, and so we get very little alteration in our estimate of the electron distribution. The uncertainty as to the Debye factor, therefore, does not introduce any appreciable error in our analysis of electron distribution. 8. It is interesting to see whether any evidence can be obtained as to whether a valency electron has been trans- ferred from one atom to the other or not. This may be put in another way : can we tell from the form of the F curves in fig. 1 whether their maxima are at 10 and 18 or at 11 and 17 respectively? It appears impossible to do this ; and, when we come to consider the problem more closely, it seems that crystal analysis must be pushed to a far greater degree of refinement before it can settle the point. If all the electrons were grouped close to the atomic centres, and if the transference of an electron meant that one electron passed from the Na group to the CI group, then a solution along the lines of that attempted by Debye and Scherrer for LiF might be possible. The electron distributions we find extend, on the other hand, right through the volume of the crystal. The distance between Na and CI centres is 2*81 A, and we find electron distributions 1 A from the centre in sodium and 1*8 A from the centre in chlorine. If the Distribution of Electrons in Na and CI Atoms. 449 valency electron is transferred from the outer region of one atom to that of the other, it will still be in the region between the two atoms for the greater part of the time, since each atom touches six neighbours, and the difference in the diffraction effects will be exceedingly small. It is for this reason that we think Debye and Scnerrer's results for LiF, which were not absolute measurements such as the above, were not adequate to decide whether the transference of a valency electron has taken place. We have assumed that the atoms are ionized in calculating our distribution curves. If, on the other hand, we had assigned 11 electrons to sodium and 17 to chlorine, we should have obtained curves of much the same shape but, with an additional electron in the outermost shells of sodium and one less in those of chlorine. 9. Summary. — We have attempted to analyse the distri- bution of electrons in the atoms of sodium and chlorine by means of our experiments on the diffraction of X-rays by these atoms. The results of the analysis are shown in figs. 3 and 4. The principal source of er.ror in our conclusions appears to be our ignorance as to the part played by "extinction" in affecting the intensity of X-ray spectra. The distributions of the electrons are deduced from the F curves (fig. 1). The most important parts of these curves are the initial regions at small angles, for errors made in absolute values in this region alter very considerably the deductions as to electron distribution. The exact form of the curve at large angles is of much less interest. Now, it is in this initial region, corresponding to strong reflexions such as (100) r (110), (222), that extinction is so uncertain a factor. Until the question of extinction is satisfactorily dealt with, the results cannot be regarded as soundly established. If our results are even approximately correct, they prove an important point. There cannot be, either in sodium or chlorine, an outer " shell " containing a group of eight electrons, or eight electrons describing orbits lying on an outer sphere. Such an arrangement would give a diffraction curve which could not be reconciled with the experimental results. Eight electrons revolving in circular orbits of the same radius would give the same diffraction curve as eight electrons on a spherical shell, and are equally inadmissible. On the other hand, it does seem possible that a combination of circular and elliptical orbits will give F curves agreeing with the observations. Phil Mag. S. 6. Vol. 44. No. 261. Sept. 1922. 2 G [ 450 ] XLIV. On the Partition of Energy. By C. Gr. Darwin, M.A., L.M.S., Fellow and Lecturer in Christ's College, Camb., and R. H. Fowler, M.A., Fellow and Lecturer in Trinity College, Camb.* § 1. Introduction. AN important branch of atomic theory is the study of the way in which energy is partitioned among an assembly of a large number of systems — molecules, Planck vibrators, etc. This study is based on the use of the principles of probability which show that one type of arrangement is much more common than any other. The most usual method is to obtain an expression for the probability of any state described statistically and then to make this probability a maximum. This always involves a use of Stirling's approxi- mation for factorials, which in many cases is illegitimate at first sight, and though it is possible to justify it subsequently, this justification is quite troublesome. It is also usually required to find the relation of the partition to the temperature rather than to the total energy of the assembly, and this is done by means of Boltzmann's theorem relating entropy to probability- — a process entailing the same unjustified approxi- mations. The object of the present paper is to show that these calculations can all be much simplified by examining the average state of the assembly instead of its most probable state. The two are actually the same, but whereas the most probable state is only found by the use of Stirling's formula, the average state can be found rigorously by the help of the multinomial theorem, together with a certain not very difficult theorem in the theory of the complex variable. By this process it is possible to evaluate the average energy of any group in the assembly, and hence to deduce the relation of the partition to temperature, without the intermediary of entropy. The temperature here is measured on a special scale, which can be most simply related to the absolute scale by the use of the theorem of equipartition, and we shall also establish the same relationship directly by connecting it with the scale of a gas thermometer. Throughout the paper the analysis is presented with some attempt at rigour, but it will be found that apart from this rigour it is exceedingly easy to apply the method of calculation. Most of the results are not * Communicated by the Authors. On the Partition of Eneryy. 451 new ; it is the point of view and the method which, we think, differ from previous treatments No discussion of the question of' partition would be com- plete without consideration of its relation to thermodynamic principles. We shall leave this view of the subject to a future paper ; for the iucreased light thrown on the statistical nature of entropy raises many interesting points which could not be discussed here properly without making the present work run to inordinate length. § 2. Statistical Principles and Weiyht. Before proceeding to the problem it will be well to review, in general outline, the principles of the theory of the partition of energy, though we have nothing new to say in this connexiou. We shall be concerned with collections of molecules, Planck vibrators, etc. — each individual unit will be called a system, and we shall call the whole collection an ■assembly. We shall be dealing mainly with assemblies com- posed of groups of systems, the individuals in each group being identical in nature. In order to make the problem definite it is necessary to assume that each system has a definite assignable energy, and yet can interact with the others. This requires that the time of interactions, during which there will be energy which cannot be assigned to a definite single system, is negligibly small compared with the time during which each system describes its own motion. For such an assembly we are to calculate various average properties of its state, when it describes its natural motion according to whatever laws it may obey. There will at any rate be an energy integral, and we have therefore to calcu- late these averages subject to the condition of constant energy. To determine the basis on which these averages are to be calculated we are to apply the principles of proba- bility ; and the calculation of itself falls into two stages, the prior and the statistical. The prior stage aims at establishing what are the states which are to be taken as of equal probability. In the statistical stage we have simply to enumerate the states specified in the prior stage, allow for the fact that the systems are macroscopically indistin- guishable, and evaluate the averages taken over these states. It is not here our purpose to enter into a full discussion of the fundamental questions that arise in connexion with the determination of what states ought to be taken as equally 2 G2 452 Messrs. C. Gr. Darwin and R. H. Fowler on probable. It will suffice to recall that for assemblies obeying the laws o£ classical mechanics the theorem of Liouville shows that the elements of equal probability may be taken to be equal elements of volume in Gibbs' " phase space." It follows out of this, for example, in the case of an assembly of a number of identical systems — say simple free mole- cules — that the elements of equal probability can be simpli- fied down into 6-dimensional cells dq l dq 2 dq z dp l dp 2 dp z of equal extension, where q l9 q 2 , q 3 are the coordinates, and Pi> Pi, ps the conjugated momenta, of a single molecule. We shall describe this by saying that the weight of every equal element dq A ... dp z is the same, and by a slight generalization, that the weights of unequal cells are pro- portional to their 6-dimensional extension. The word weight is here used in exactly the sense of the term a priori probability, as used by Bohr and others. But when we come to the quantum theory, mechanical principles cease to hold, and we require a new basis for assigning the equally probable elements. Such a basis is provided by Ehrenfest' s * Adiabatic Hypothesis and Bohr's f Correspondence Principle. These show how the theorem of Lionville is to be extended, and allow us to assign a weight for each quantized state of a system. It is found that we must assign an equal weight to every permissible state in each quantized degree of freedom. At first sight this is a little surprising, for it would seem natural to suppose that a vibrator which could only take energy in large units would be less likely to have a unit than one which could take it in small ; but this is to confuse the two stages of the problem. It is only by the supposition of equal weights that we can obtain consistency with classical mechanics by the Corre- spondence Principle. It is customary % in assigning a definite weight to every quantized state to give it the value h, so as to bring the result to the same dimensions as those of the element dq dp in the classical case. But there is considerable advantage in reversing this, and taking the quantized weights as unity and the weight of the element in the phase space as dq dp/h ; for if this is done, the arguments about entropy are simplified by the absence of logarithms of dimensional quantities. We shall adopt this convention here, though in * Ehrenfest, Proc. Acad. Amst. xvi. p. 591 ; Phil. Maof. xxxiii. p. 500 (1917), etc. t Bohr, " The Quantum Theory of Line Spectra," Dan. Acad. iv. p. 1 (1918). % Ehrenfest & Trkal, Proc. Amst. Acad. Sc. xxiii. p. 162. See in par- ticular p. 165 and Additional Notes, No. 1. the Partition of Energy. 453 all our results it is immaterial — indeed, until such questions as dissociation are considered it makes no difference to adopt different conventions for different types of system. The convention has the advantage of shortening a oood manv formula? and freeing them from factors which are without effect on the final results. An exception to the above rule for assigning- weights to quantized motions occurs in the case of degenerate si/stems, where there are two degrees of freedom possessing the same or commensurable frequencies. In this case there is only one quantum number, and the state of the system is partly arbitrary. Bohr * shows that the rational generalization is to assign to such a state a weight factor which can be evaluated by treating the system as the limit of a non- degenerate system, and quantizing it according to any pair of variables in which it is possible to do so. The number of the permissible states which possess the same total quantum number will give the weight of the state. A corresponding- rule holds for systems degenerate in three or more degrees of freedom. The meaning of weight can perhaps be made clearer by considering its introduction the other way round — beginning with an assembly of simple quantized systems of various frequencies. Griven the energy, there is a definite number of possible states, which are fully specified by the energy assigned to each system. We then make the hypothesis that it is right to assign an equal probability to each such state in the calculation of averages. This is now the fundamental postulate. The generalization to degenerate systems goes as before, by introducing weight factors. Finally, passing over to mechanical systems, such as free molecules, we are led by an appeal to the converse of the Correspondence Principle to attach weight dq x . . . dp^/h 3 to each 6-dimen- sional cell which specifies completely the state of a single molecule. The second, statistical, half of the problem consists in enumerating the various complexions possible to the assembly. By a complexion we mean every arrangement of the assembly, in which we are supposed to be able to distinguish the in- dividuality of the separate systems. We count up the total number of complexions which conform to any specified statistical state of the assembly, and attach to each the appropriate weight factor. Thus the probability of this state is the ratio of the number of its weighted complexions to the * Bohr, loc. cit. p. 26. 454 Messrs. C. G. Darwin and R. H. Fowler on total number of all possible weighted complexions. This part of the problem depends on the nature of the particular assembly considered, and so must be treated separately in each case. We start in § 3 with a problem which concerns not the partition of energy, but the distribution of molecules in a volume. It illustrates the method in its simplest aspect and has the advantage of being purely algebraic. Next, in § 4, we take the distribution of energy among a set of similar Planck vibrators, which is again a purely algebraic process, and then proceed in § 5 to introduce the main theme of this paper by dealing with the partition of energy between two sets of Planck vibrators of different type. This is most conveniently treated by using the complex variable, and in § 6 there is a discussion of the required theorem. The par- tition of itself introduces the temperature, and in § 7 the special scale is compared with the absolute. In §§ 9, 10, 11 the partition law is generalized to more complicated types of system, such as the quantized rotations of molecules. In §§ 12, 13 the method is extended so as to deal with the free motion of monatomic molecules, intermixed with vibrators. The work leads to a rather neat method of establishing the Maxwell distribution law. § 3. The Distribution of Molecules in Space. The first example we shall take is not one of a partition of energy, but of the distribution of small molecules in a vessel. It illustrates in its simplest form the averaging process, and has the advantage of depending only on elementary algebra. Let there be M molecules, and divide the vessel into m cells of equal or unequal volumes r l5 v 2 ... v m , which may each be as large or as small as we like. Then Vi + V 2 + ... +v m = V (3-1) By well-known arguments which we need not consider, it follows that any one molecule is as likely to be in any element of volume as in any other equal one. So by a slight ex- tension of the idea of weight we attach weights v h v 2 , ... v m to the cells. To specify the statistical state we say that the first cells has a x molecules, the second a 2 , and so on. Then a 1 + a 2 -\- ... +a m = M (3*2) By the theory of permutations the number of complexions the Partition of Energy, 455 which conform to the specification is M! a x ! a 2 ! . . . a m I ' and each of these must be weighted with a factor u l u % • • • ' »i The total number of all the weighted complexions is ai ! a 2 ! . . . = (, 1+! . 2+ ...) M = VM, by the multinominal theorem. This could have been deduced at once by working direct with probabilities v r /Y instead of weights v y , but the argument has been given in detail to illustrate the method for more complicated cases. We next find the average value of a r . This is given by M' Qa r = % a a r , vfivf* ... . Ui 1 «2 ' • • • To sum this expression we only have to cancel a r with the first factor in a r ! in the denominator, and then it is seen to be equal to Mv r (v l + v 2 + ...yi-\ and so, as is implicit in our assumptions, a, = Mt V /V (3'3) But we can now go further and find the range over which a r will be likely to fluctuate. This is estimated by averaging the square of the difference of a r from its mean value. We shall throughout this paper describe such a mean square departure as the fluctuation of the corresponding quantity. Thus the fluctuation of a r is (ar — a r ) 2 . Now (a r — a r ) 2 = a r (a r — 1) + a r — 2a ~d r -\-~a r 2 , and averaging the separate terms by the multinominal theorem, we have (a r a r ) — y 2 + y V ' V + \ V / = ^(l-v) = -( 1 -M> • • • ^ This result represents the fluctuation however large or small 456 Messrs. C. G. Darwin and R. H. Fowler on v r may be. In all cases we have the result that (a r — a r ) 2 is less than a~ r , and therefore that the average departure of a r from ~a r is of order (o^)*. We can also interpret this fact by saying that departures of a r from li r which are much greater than (a7-)* will be relatively rare ; as M is large and (or)i small compared with oT r , this is precisely equivalent to saying that the possession of the average value of a r is a normal property of the assembly in the sense used by Jeans *« We have thus a simple and complete proof that uniform density is a normal property of this assembly. § 4. The Distribution of Energy among a Set of Planck Vibrators. Another case where the treatment can be almost entirely algebraic is that of the partition of energy among a set of Planck vibrators which all have the same frequency. Let e be the unit of energy so that every vibrator can have any multiple of e. As we saw in § 2, the weight attached to every state is to be taken as unity. Let there be M vibrators and let there be Pe of energy (P is an integer) to be partitioned among them. To specify a statistical state, let a be the number of vibrators with no energy, a x with e, a 2 with 2e, etc. Then we have a + a 1 + a 2 + a 3 + ... = 'M., .... (4'1) a 1 + 2a 2 + 3a 3 +... = P, .... (4'2) and anv set of a's which satisfy these equations corresponds to a possible state of the assembly. By the principles of § 2 each of the complexions will have unit weight. Now count up the number of complexions corresponding to the speci- fication. By considering the various permutations of the vibrators, it is seen to be M! a \ ail a 2 l (4-3) We must next find C the total number of all possible complexions. Let X a denote summation over all possible values of the a's which satisfy (4*1) and (4*2). Then a ! aj : a 2 l ... Consider the infinite series (i+s + si +5 3 + ; >)M - * Jeans, ' Dynamical Theory of Gases/ passim. the Partition of Energy. 457 ■expanded by the multinominal theorem. The typical term is a ! a x ! a 2 \ ...~ where the as take any values consistent with (4*1). Then if we pick out the coefficient of z F , we have the sum of all the expressions for which the a's satisfy both (4*1) and (4*2). Observe that we may take the whole infinite series because the later terms are automatically excluded. Now this will be the coefficient of X s in (1—z) ~ M , and so (M + P-l)! a . 4 . °- (M_i)!P! ' ^** J which is the ordinary expression for the number of homo- geneous products as formerly used by Planck *. We next evaluate the average of a r ; M! a ! a l ! a 2 1 . . . -MS, ( M -^ ! a ! a l ! a 2 I . . . where X a > denotes summation over all values satisfying «o' + «i / + « 2 / + «3 / --- =M — 1, a/ + 2a 2 / 4 3a 3 / ... = P — r. The sum is thus M (MtP-r-2)! (M-2)!(P-r)!' and we have q-M(M n ( M + P-r-2)! P! This is exact, and holds for all values of r ; now r can have any value up to P and the majority of the a r 's will be zero. The ordinary method of proof applies Stirling's formula for a r ! to these zero values. In the important case where both M and P are large, it will be only necessary to consider values of r which are small compared with P. Now, if r 2 is small compared with P, P!/(P — r)! has the asymptotic value P r . Using relations of this type and also disregarding the difference between M and M — 1, we have M 2 P r "*= (M+ty+i (4 ' 5) * M. Planck in the earlier editions of his book on Radiation. 458 Messrs. C. Gr. Darwin and R. H. Fowler on The same methods give the fluctuations of a r . For (a r — a r ) 2 — a r a r — 1) -f a r — (a r ) 2 , and a process similar to the above gives 0-^?^=T) = M(M - X) The exact expression for the fluctuation can be at once put down. When M and P are taken large the leading term cuts out, and so it is necessary to carr j the approximation to the next order. If we substitute P!/(P-r)I--P^-ir(r-I)F- 1 , we find that \(Xf (Jby J — £ { l ~ (M + Pr* [2P 2 -(2?-l)MP + r*W] j . (4-6) The formula for a r can be put into a more familiar form by the substitution P = M/(^ a — 1), which gives ~a r = Me- ra (l — e- a ), . . . . (4 ! 7) and leads to a corresponding but more complicated ex- pression for the fluctuation. Here, as we shall see later, ol can be identified with the familiar e/kT., Equation (4* 6) establishes at once that the statistical state specified by (4*5) or (4* 7) is a normal property of the assembly. § 5. The Partition of Energy among two Sets of Planck Vibrators. After these preliminary examples we now apply our method to a problem which will bring out its distinctive character, that of the partition of energy in an assembly composed of different types of system. We shall consider first the simplest of such cases — an assembly consisting of a large number of Planck vibrators of two types A and B. The number of A's- is M, and the energy unit of an A is e as before. There are now also N B's with energy unit tj. To make exchanges of energy possible we have to suppose, say, that there are present a few gas molecules, but that the latter never possess any sensible amount of energy. (Later on in § 12 we develop a method by which we shall be able to include any number of such molecules in our assembly.) We also require for the purposes of the proof to assume that e and rj are com- mensurable, but it does not matter how large the numbers "may be which are required to express the ratio e/rj in its the Partition of Energy. 459 lowest terms. To avoid introducing new symbols, we may suppose that the unit of energy is so chosen that e and r\ are themselves integers without a common factor. We have already introduced the idea of weight, and seen that we must assign the weight unity to every permissible state of a linear vibrator. To calculate the number of com- plexions of the assembly of any given sort, we have merely to calculate the number of ways in which the energy may be distributed among the vibrators, subject to the given statistical specification. A simple example will make the process clear. Let there be two A's and two B's; let ^ = 2e, E = 4e. Then the possible complexions are : — {aaaa r i r a a a < ( aaaa' I aa'a'a aab aab' a'ab Wa'b' job \aa!b \aa'V bb' Here, for example, aaaa' means that there is 3e of energy on the first of the A's, e on the second, and none on the B's. Each of the fourteen complexions is, by definition, of equal weight, and is therefore to be reckoned as of equal probability in the calculation of averages. Observe how with the small amount of energy available a, good deal more goes into the smaller than into the larger quanta ; for the pair of A's have on the average ye, as against ye for the pair of B's. We pass to the general case. The statistical state of the assembly is specified by sets of numbers a r , b s where a r is the number of vibrators of type A which have energy re, and b s the number of B's with energy stj. All weights are unity and the number of complexions representing this statistical state is the number of indistinguishable ways (combinations) in which M vibrators can be divided into sets of a , a Y ... and at the same time N into sets b , b ± ... . As illustrated by the example, it is therefore given by the formula M! N! o !a l la 1 !...* a !6 l !W..." * * * ( ; In (5"1) a r and b s may have any zero or positive values consistent with the conditions 2 r a r = M, SA = N, 2 r r6a r + X s s V b s = E, . (5'2) where E is the total energy of the system — necessarily an 460 Messrs. C. G. Darwin and It. H. Fowler on integer in the units we employ. The total number C of all complexions is therefore M ! N ! G = ^ b aJa'JaTj::. b l b ± l b 2 \ ... ' * * (5 ' 3) where the summation X a ,b is to be carried out over all positive or zero values of a r and b s which satisfy (5*2). By using (5*1) and (5*3) we can at once obtain an expression for the average value, taken over all complexions, of any quantity in which we are interested. We have already studied ~a T in § 4. The main interest centres in Ea, the average energy on the A's. We have at once PF -? (2,.r £ q r )M!N! ^-^A — A*, b i 1 1 7 I 7 I 7 I • • • \p ^) The following process leads to simple integrals to express the quantities C, CEa, etc. Consider the infinite series (l + z e + z 2e + ) M expanded in powers of z by the multinominal theorem. The general term is 'Ml z ? r rea„ m a Q ! % ! ..." It follows that if we select from the expansion of (l + * e + ^ e +...) M (l + s*^* + ...)* . (5-5) the coefficient of z E , we shall obtain the sum of all possible terms such as (5*1) subject to the conditions (5*2), that is to say C Similarly, if we form the expression j^(l+s e H-2 2e ...) M 1(1-^ + ^...)*, . (5-6) the general term in the first bracket must be {% r T€a r )M\ 2 r rea r — Z , \^\ax\ and by the same reasoning the coefficient of z E in (5*6) must be CE A . Expressions (5*5) and (5*6) are easily simplified— they are respectively the Partition of Energy. 461 The latter can also be written as {-Mjrgglog (!-<«) }(l-*«)-*(l. If these expressions are now expanded in powers of z by the binomial theorem, they give a sum of products of factorials which are, of course, the " homogeneous product " expressions used by Planck. It is possible to approximate to these by a legitimate use of Stirling's theorem and to replace the sums by integrals without much difficulty. It would, indeed, have been possibly to start from these expressions, but we have not done so because in the general case to be discussed later that method would not be available. To make further progress* by a method of general utility, we discard Stirling's theorem and express these coefficients of z® by contour integrals taken round a circle y with centre at the origin and radius less than unity. By well-known theorems on in- tegration * we have at once c i r dz i . = 2mUz E+1 (l-z e ) M (l--z r i)K> ' ' (5 ' 71) d -Ms-T-logd-s 6 ) — 1 C d~ ~ dz & K J CEA= 2^-J y ^i-(1_^M (1 _^ '• ' ( 5 * 72 > AVe can no longer hope for the single-term formulae of §§ 3 ; 4. But (5-71) (5-72) are exact, and when M, N, E are all large in any definite fixed ratios, we can make use of the method of steepest descents to obtain simple adequate approxi- mations. The method is very powerful and can be applied in a great number of cases without difficulty. Moreover, it is comparatively easy to use it with mathematical rigour if that is desired ; and thus the somewhat clumsy calculations in the usual proofs of partition theorems are entirely avoided. In general terms the process is this. Consider the in- tegrand on the real positive axis. It becomes infinite at z = and again at z=l, and somewhere between at z = § there is a minimum which is easily shown to be unique. Take as the contour the circle with centre at the origin and radius S passing through this minimum. Then we find that, for * For those not familiar with these theorems we may remark that 5— -. 1 2 r <7z=0 when r is any integer other than — 1, while r> — ' 1 — =1*;. these equations at once give (5'71) and (5 - 72). 462 Messrs. C. G. Darwin and R. H. Fowler on values of z on the contour, z = § corresponds to a strong maxi- mum, and when M, N, E are large, such a strong maximum that practically the whole value of the integral is contributed by the contour in the neighbourhood of this point. Hence in the integrals it is legitimate to substitute the value at this point tor any factors which do not themselves show strong maxima here or, elsewhere. On this general principle we can remove the term — Wz -j-^ log (1— z e ) from under the integral sign, provided that z is given the value S determined by the maximum condition. The part of the integrand in- volving the large numbers M, N, E, determines the value of ■■& as being the unique real positive fractional root of the equation j- |^- E (1-0 6 )" M (1-^)" N }=O. That is, 3 satisfies the equation : The remaining integrands in C and CEa are identical, and we therefore have El=-M^log(l-n -A n If a similar process is carried out for the B's, we have s^plrr ( 5-91 ) in agreement with the necessary relation Ea + Eb = E. Equations (5*9) (5*91) determine the partition of energy and take their familiar form if we replace S by e~ 1/kT . We shall return to this point later. § 6. Application of the Method of Steepest Descents. After this sketch it will be well to establish the validity of the ar uments used. This section is put in for mathematical completeness, and is not concerned at all with physical questions. We treat of a more general case than that of § 5. the Partition of Energy. 463 Arguments of tins type — asymptotic expansions by steepest descents — are, of course, well known in pure mathematics. Consider a contour integral of the form isj/WW-)]"* .... (6-1) subject to the following conditions : — (i.) <\>{z) is an analytic function of z, which can be ex- panded in a series of ascending powers of z. (ii.) This series starts with some negative power. (iii.) Every coefficient is real and positive. (iv.) Its circle of convergence is of radius unity. (This condition is quite unessential to the mathematics, but makes the statement simpler, and is physically true.) (v.) The powers that occur in the series cannot all be put in the form a + /3r- where a and ft are any given integers and r takes all integer values. (vi.) F(r) is an analytic function with no poles in the unit circle, except perhaps at the origin, (vii.) 7 is a closed contour going once counterclockwise round the origin. The problem is to obtain the asymptotic value as E tends through integer values to infinity. We shall first study the properties of <j>(z). Considering real values, it must have one and only one minimum between and 1. For it is continuous and tends to + co at both and 1, and so must have at least one minimum between. Further, to find minima we differentiate, and then all the negative powers will have negative coefficients and all the positive positive. It follows that minima are given where two curves cut, one of which decreases steadily between and 1, while the other increases steadily. These curves can only cut in one point, and so there is only one minimum. Xext, for the complex values, consider a circle of any radius r less than 1. As the modulus of a sum is never greater than the sum of the moduli, it follows that at no point on this circle can \<j>(z)\ be greater than <f>(r). Moreover, it can only equal 4>(r) provided that condition (v.) is broken, and in that case there will be /3 points at each of which \<j>(z)\ = (j>(r). We can thus see that on account of the large exponent it is only the part of <y near the real axis that contributes effec- tively to the asymptotic value of the integral. This suggests 464 Messrs. 0, Gr. Darwin and R. H. Fowler on the substitution z = re iQI , with a as the new variable of integra- tion. Expanding near the real axis we have [<K*)] E = W>0)] E exp. {iaEr<£7</>+ 0(E« 3 )}. (6'11) This function contains a periodic term of high frequency which cuts down the contributions for small values of a, so that the value of the integral will in general depend on more distant parts of the contour than those to which this approxi- mation will apply. But if we choose for r the special value 3 corresponding to the unique minimum <$>' = 0, then the oscillating term in the exponential vanishes and the contribu- tions for small values of a dominate the whole integral. For this special value of r the exponential becomes exp.{-Pa 2 d 2 (/)' , /</) + 0(Ea 3 )}, . . (6-12) and by (iii.) <j> f/ >0. We see at once that we can suppose that Wa ranges effectively over all values from — oo to + oo while all other terms, such as a,..., Ea 3 , ... remain small. We then obtain for (6*1) on putting z = §e ia the asymptotic expression — OG^P f {F(S) + ; a 3F'(3) + 6>0 2 ) + O(E« 3 )}6-PaW70^ For most purposes the first term in the expansion will suffice, but if the precise values of the fluctuations are re- quired, the second also is necessary. As it is in general rather complicated, we shall content ourselves here with pointing out its order of magnitude. On carrying through the necessary calculations we find The argument of F and <£ is everywhere 3 ; the term {§, </>, F} denotes a complicated expression of ■&, <£ and its first four derivatives, and F and its first two derivatives, but is independent of E. If condition (v.) is dropped, we shall have /3 equal maxima arranged round the circle 7, and, pro- vided F has the same value at each of them, the integral will have a value equal to (6'2) multiplied by J3. Now consider the problem of § 5, to which our work applies immediately with We may suppose that E tends to infinity and also M and N the Partition of Energy. 4G5 in such a way that M/E and N/E are constant. This func- tion satisfies all the conditions of this section — the fact that it is in general many-valued is irrelevant, for we are only concerned with that particular branch which is real when z is real and 0<^<1 and this branch is one-valued in the unit circle. The unique minimum S is determined by the equa- tion cj>'=0 or Me N^ E =s=zzi + $=v-l ( fi "4) The value of the integral (5*71) is then by {6'2) (omitting the second term the form of which is only required in calculating fluctuations), s-B(i-y)-M(i-s*)-g If, contrary to hypothesis, we had taken e and rj as having a common factor /3, condition (v.) would have been violated. In this case C would be /3 times as great as before, but so would CE A , so that E A and all other averages would be unaffected. The use of (6'2) to evaluate CE A (5 - 72) etc. leads at once to expressions similar to (6*5), and so to the results given formally in the last section and to others to be given later. As we shall see, ■& is the temperature measured on a special scale, and there is great advantage in regarding 3, rather than E, as the independent variable which determines the state of the assembty. If this is done the expression ~E$ 2 (f>"/(f) can be put into simpler form. For it is easily verified that Eayv<ft= ^* * + v (a- e -i) 2 ' (&-"-i) a ' =*SIM»>=*§' ■ • ( 6 " 6 ) if E is regarded as a function of S, given by (5'S). It may be remarked that the constant occurrence of the operator $d/d§ suggests a change of variable to log 3. Though this has some advantages we have not adopted it, partly because it makes the initial argument about the multinomial theorem a little harder to follow and complicates the contour of the integration, and also because log $ is not itself the absolute temperature — if it had been, the physical simplicity might have outweighed the other objections — but only a quantity proportional to its reciprocal. Phil. Mag. S. 6. Vol. 44. No. 261. Sept. 1922. 2 H 466 Messrs. 0. G. Darwin and R. H. Fowler on § 7. The Meaning of '&. Returning to the subject of § 5, we see that the partition of energy between the A's and the B's depends on a para- meter which can be determined in terms of" the energ}- by means of the equation (5*8). It is fairly evident that 3 is connected with the temperature, though it requires more general considerations to prove this properly. But if we assume this connexion and are content to replace the thermo- dynamic definition of absolute temperature by one based on the law of equipartition of energy for systems obeying the laws of classical mechanics, then we can at once identify the meaning of $. For let us suppose that the B's are vibrators of very low frequency. They will then obey the classical laws, and the average energy of each will be kT. But Tit v — — £T ^Js-i-i.-iogCi/*)-*- 1 ' which shows that *=«-^ T (7-1) Substituting this in (5*9), we obtain the well-known form Me E A = e e/kT Observe that while rj is tending to zero there need be no difficulty about the condition that rj and e are to be com- mensurable. We shall later return to this question of temperature and establish it for much more general types o£ system. § 8. Other mean values. Exactly the same methods can be applied to evaluate any other mean value besides E A . For example : — p - _ v jvn ni Ua r-A,» a r^j ai !. ... 6 ! b, !....' _ (M-l)l N! - M ^WUi'!.... MM-.:..:' * ( } summed over all zero and positive values such that X r a r ' = M-l, 2A=N, % r r*a r ' + 2,sf,b=$-re. Applying the multinomial theorem and reducing the the Partition of Energy. 467 expression to a complex integral, we have which, by virtue of the value of C from (5' 71) and the argument of § 6, at once yields a;=Mr(i-y), =M*-«/**(l-e-*/* T ) J . . . . (8-3) which is the formula of § 4 over again, the presence of the B's being immaterial. When we come to evaluate fluctuations the matter is a little more complicated, because the leading terms cut out, and so the second term of the asymptotic expansion will in general play a part. For example, consider the fluctuations of a_ : ( a r- a r) 2 = a X a r- 1 ) + ^-<X) 2 ' By arguments exactly similar to those above, we have C«„( a.-l^MCM-D^j^Tw (1_/)N- 2 (1_^)*" and so by (6-2) «>,- 1) =M(M-ip 2 " £ (l-y) 2 -U+ 0(1/E) }. Thus the fluctuation is ^_M3 2r6 (l-3 e ) 2 + 0(M 2 /E). This is sufficient to show that the possession of ~a~ r (8'3) is a normal property of the assembly. The complete calculation of the O-term is rather com- plicated ; the result is given at the end of this section. But a great simplification arises if we suppose that there are many more B's than A's, while E is so adjusted that $ the temperature is unchanged. In this case the term 0(M 2 /E) becomes small and may be neglected. We shall describe this case by saying that A is in a bath of temperature S. Then, provided this is so, we have (^EJ=a r Q--aJ-M.) (8-4) A much more important quantity is the fluctuation of E A . This is found by evaluating E A 2 . Now, just as CE A was given by operating with zd/dz on the first factor in 2R2 468 Messrs. C- G. Darwin and R. H. Fowler on (1 — 2 6 )~ M (1— ^) -N , so CE A 2 is easily seen to be given by operating with (~^) in the same way. Thus ce?= ^[% { («£) Wr- } (i-^)-*. If we again suppose an infinite bath of temperature $, we can omit the second term of the asymptotic expansion (6'2) and obtain e>(i_*O m {(^) 2 (i-*T m }, =(i-y) M ^{A(i-*r*b — <^7 and so the fluctuation is (E A -E A )*=E7-(E A )* -iKP ^ (8-5) J This is a result of which Einstein * made use in his work on fluctuations of radiation. It should be emphasized that these results are only accurate in a temperature bath, and not when the number of systems A is a finite fraction of the assembly. In all cases (6*2) shows that the possession of E A is a norma! property of the assembly. If we work out exactly the second terms in the asymptotic formulse of § 6 and apply them to the fluctuations of a r and E A we find (a r -a,)'-a r [l- g |l+ ^ E /cft J J' < 8 6 > ^^-^^^fV^f) • • • <*?> Formula (4*6) above is a special case of (8*6). * A. Einstein, Phys. Zeitschr. vol. x. p. 185 (1909). the Partition of Energy. 469 Finally it is of some interest to point out that we can obtain a formula for (Ea— "Ea) & of general validity. We have in fact (E A -E A ) 2 * = 1.3...'(2s-l){(E A -E A ) 2 }*, . ($•$) where (E A — E A ) 2 is given by (8*7). We retain of course only the highest order term *, which is thus 0(E A )*. § 9. Generalization to any number of types of system, and to systems of any quantized character. It is clear that the present method of treating partitions is of a much more general character than has so far been exhibited. Consider an assembly composed of two types, A and B, of quantized systems more complicated than Planck vibrators. We suppose generally that the systems of type A, M in number, can take energies to the extents e , e 1} e 2 , ..., and these states have weight factors po^Pi->I ) 2^ ...in conformity with the discussion in § 2. Similarly, the B's, N in number, can take energies rj , 77^ tj 2i ... with weights q , g^ q 2 , We have to suppose that it is possible to determine a basal unit of energy such that all the e's and n's can be expressed as integers. Further, it simplifies the work if we suppose that there is no factor common to all of them. Proceeding exactly as before, we set down the weighted number of com- plexions which correspond to the specification that, of the A's, a r have energy e r ; of the B's, b s have energy ij s . This number is a ,. , wv • • • • b , ; ?«v . . . ., • (9-i) a Q i a 1 i . . . o ! o 1 : . . . and the a's and 6's are able to take all values consistent with \a=U, 2A = N, S r ^+S^A=E. . (9-2) Now form the functions f(z)=p a z e « + /h ^+p.^ +...., . . . (9-3) 9 (z)=q z"° + q^+q,^+ (9-31) These will be called the partition functions f of the types of * Cf. Gibbs' < Statistical Mechanics,' p. 78. But (8*8) is generally valid, while Gibbs' formulae really refer only to a group of systems in a temperature bath. t They are practically the " Zustandsumme '' of Planck, ' Radiation Theory/ p. 127. 470 Messrs. C. G. Darwin and R. H. Fowler on system A and B. The application of the multinomial theorem then leads to the consideration o£ the expression [/(*)] M M*)f, and pursuing exactly the same course as in § 5, we find C=2-y^ 1 [/«] M [^)] N , • • • (9-4) Assume for the moment that we can choose a <\>(z) con- forming to the requirements of § 6. The whole calculation then goes on as before. The radius of the circle to be taken as contour is given by the equation E = M^log/(3)+N^ log </(£). . . (9-6) This equation has one and only one root. We thus can at once put down , *> \ . . . (9-7) =M3 J log /(d). ) In exactly the same way we have £=Mp,**/X»). (9-8) and we can also verify that in the case of an infinite bath the fluctuations are again given by (8*4), (8*5), and that equation (6'6) is still true. The exact forms of the fluctua- tions {8'$)i (8*7) are also valid if we replace re by e r . We have now to examine whether <j>(z) can be properly chosen. It is natural to take <f>(z)= Z -H/W Mm Um m - ■ ■ ■ (9-8 1 ) By its definition it must satisfy (i.). For (ii.) to be true, we must have E>Me + N>/o, which is the trivial condition that there must be enough energy to provide each system with the least amount Of energy it is permitted to have. Condition (iv.) does not appear at first sight inevitable, but must follow from Bohr's Correspondence Principle *, for the convergence of the series f(z) and g(z) depends on their later terms — that is, those of * Bohr, loc. cit. tlie Partition of Energy. 471 large quantum numbers. Condition (v.) is satisfied if not all the e's and v's have a common factor. There remains (ill/), and here there are trivial analytical difficulties when, as in general, M/E and N/E are fractional. It is, however, easy to generalize § 6 by replacing [</>(c )] E h J and letting E, M, N all tend to infinity independently. Condition (iii.) is then satisfied, as can be seen by multiplying out, and so all the conditions are satisfied, and the final results stated above are unaffected. Finally, we may observe that all our results can be extended at once to an assembly containing any number of types of system. If there are M c systems of type C, for which the partition function is/ c (3), then B.=lW^-log/.(»). where 3 is determined by E=S M^log/ c ($). The formal validity of the proof will require all the quantities e c to be commensurable. It will be shown in § 12 how this restriction may be removed. § 10. Vibrators of two and three degrees of freedom. As a first example we take a set of vibrators each of which is free to vibrate in a plane under a central force proportional to the distance. The sequence of energies is again 0, e, 2e, ..., but the weights are no longer unity, as the system is degenerate. Following the principle laid down in § 2, we may evaluate the weights by treating the system as non-de- generate and counting the number of different motions which have the same total quantum number. Now we can quantize the plane vibrators in directions x and y, and as an example for the case 2e, we have three alternatives (2e, 0), (e, e), (0, 2e). This is easily generalized, and gives to re the weight r-f-1. The partition function for these vibrators is thus /(c) = l + 2* + 3; 2 +4s 3 +..., 472 Messrs. C. G. Darwin and R. H. Fowler on From the general theorem (9*7) we at once have so that such vibrators have just twice as much energy as the line vibrators. In exactly the same way we can treat the case of three dimensions. To illustrate the weights we again take the case of 2e and quantize the svstem in x, y, z. There are six alternatives (2e, 0, 0), (0/2e, 0), (0, 6, 2e), (e, €, 0), (e, 0, e), (0, e, e). The general form for re is J(r+ l)'(f + 2). The partition function is now = (l-e)" 3 5 which leads at once to the expected result § 11. Rotating Molecules. Another interesting example to which the calculations at once apply is that part of the specific heat of a gas due to the rotations of the molecules. Various writers * have quantized the motions of a rigid body, and it is found that the system has at most two instead of three periods, so that it is partly degenerate. We may consider for simplicity a diatomic molecule. Then, on account of the small moment of inertia about the line of centres, the third degree of freedom may be omitted altogether — its quantum of energy is too large. A simple calculation then leads to energies of rotation e r given by ^=*5r 2 > c*d where I is the moment of inertia about a transverse axis, which we shall assume to be independent of r. This is a degenerate system, and considerations of the number of cases which occur if it is quantized for the two degrees shows that the weight to be attached is 2r-f 1. This is on the principles suggested by Bohr f with a simplifying modifica- tion; for Bohr had to suppose that certain quantized motions were excluded for other reasons which are not operative * Among others, Ehrenfest, Verh. Deutsch. Phys. Ges. xv. p. 451 (1913). Epstein, Phys. Zeitschr. xx. p. 289 (1919;. F. Reiche, Ann. der Physik, liv. p. 421 (1917). f Loc. cit. p. 26. the Partition of Energy. 473 here *. There can, we think, be no question as to the correct- ness of the weight 2r+l, bat most recent writers have used a factor?*; our formula for the specific heat has therefore a rather different value. We may now apply our general formulae to this case with p r =2r+l, e=rh, 6=^. . . (11-2) Then /(^) = l + 3S 6 + 5^ fc + 7S 9fc '+..., . . (H-3) E A = M$J^log/(a) (11-31) The contribution of the rotations to the molecular specific heat, C rot , is dEJdT, where M must be taken as the number of molecules in one gramme-molecule of gas. Thus, using (7*1), we have M.k /. d 2 5 (^)log/W, • • (H-4) Crot (log&J s and M£ = R, the usual gas constant. If we write *-k5et' (11 ' 5) Crot^R^ — logil + Ze-' + Oe-^+le- 9 "-)-...). (11-51) then da* Equation (11*5) shows that, when T— >co, a— >0. It can be shown by the application of standard theorems on series f that when <r->0, C, *->B, (11-52) which is the correct limiting value as required by classical dynamics. In the general case of any body we have three degrees of rotational freedom, the motion is simply degenerate J, and the energy enters as a sum of square numbers multi- plying two units of energy. The motion of the axis of symmetry and the motion about the axis of symmetry are not independent, and it is impossible therefore for the parti- tion function to split up into the product of two partition functions which represent the separate contributions of the two motions. The result is a double series of the same general type as (11*3). * Assuming that no extraneous considerations rule out any of these states. t JBromwich, Infinite Series, p. 132. The theorem is due to Cesaro. X Epstein, Phys. Zeit xx. p. 289 (lyl9). 474 Messrs. C. G. Darwin and R. H. Fowler on It does not appear profitable to examine these expressions further here, since the agreement with experiment is not very good at all temperatures. It is to be presumed that the assumption of constant moments of inertia is at fault, and this is supported by some of the evidence from band spectra ; further, it is probable that the case of no rotation must be excluded, involving the omission of the first term in the partition functions. The discussion of the practical applications of these formulae cannot be entered into here. § 12. Assemblies containing free molecules. The problems we have so far discussed have all possessed the distinguishing characteristic that the temperature is the only independent variable. As soon as we treat of free molecules this is no longer the case, for now the volume must be another independent variable. Nevertheless, as we shall see, the same methods of calculation are available. The partition is no longer represented exactly by integrals, as it was for the quantized motions, but from the nature of the case some form of limiting process is required. The free molecules cannot of course be regarded as the limit of three-dimensional vibrators of low frequency, for they have no potential energy to share in the partition. We must proceed by the method common to most discussions of the distribution laws of classical assemblies — divide up into cells the six-dimensional space in which the state of any molecule is represented, associate with each cell a certain constant value of the energy, and in the limit make all the dimensions of all the cells tend to zero *. We take an assembly composed of M systems of the type A of § 9 and P free-moving monatomic molecules of mass m and of small size, the whole enclosed in a vessel of volume V. The energy of the molecules is solely their energy of trans- lation ; they are supposed to obey the laws of classical mechanics (except during their collisions with the A's). In order to specify the state of the assembly, we*fcake a six- dimensional space of co-ordinates q u q 2 ,...p 3 , the three rectangular co-ordinates and momenta of a molecule in the vessel. We divide up this space into small cells, 1, 2, 3, ..., t ..., of extensions (dq l ...dp z ) u which may or may not be * That the limit of the distribution laws worked out for the cells is the true distribution law for the actual assembly is an assumption implicit in all such discussions. the Partition of Energy. 475 equal. Then by the principles of § 2 the weight factor for the tth cell is 8t= («** •» <fo\ .... ( i2-l) provided of course that the cell is relevant to our assembly. Only those cells have a weight for which the q } s lie in the vessel; but the //s may range over all values from -co to + co, for the method of summation will automatically exclude values which could not be allowed. Associated with the tth cell there is energy given by r ( =^0'i 2 +^ 2 +ps 2 ). . • • (12-iD The state of the molecules in the assembly is specified by the numbers c l5 c 2 , ... of molecules in cells 1, 2, The specification of the A's is as before. The number of weighted complexions corresponding to the specification is then Mf Pi " '— i>o> gl ... > t V^V., . (12-12) . a ! fll !../ u ^ •••c 1 U' 2 ! where Z r a r = M, $ t c t = F, 2,.a r e r + W* = E - . (12-13) In proceeding thus we are constructing an artificial assembly in which the energy is taken to have the same value f f in all parts of the tth cell, and in which all the f's and all the e's can be expressed as multiples of some basal unit, without a factor common to thern all. This assembly can be made to resemble the real one to any standard of approximation required. For such an artificial assembly w r e can make use of the whole of our machinery. The results all depend on integrals such as ■*T [««)]*[*«]*, • • a 2 " 2 ) where the partition function of the artificial molecules is fcO)=2A/<, (1*21) and the formulae of § 8 follow at once for E c , (E c — E c ) 2 , c t and (<' t — c t ) 2 . These results give completely the exact partition laws for any artificial assembly of the type considered. To obtain the actual distribution law for the real assembly, we must make all the dimensions of all the cells tend to zero, and obtain the limit of the partition function. Now, by the 476 Messrs. G. Gr. Darwin and R. H. Fowler on definition of an integral, in the limit A($)-»H($), where H(*)= ^e-^W^d* ... d n . (12-3) The integration is over the volume V and over all values of the p's from -co to + co . This gives at once (27nn)3/*V In the formulae the functions dh/dS and d 2 h/d§ 2 also occur, and it is easily shown directly that their limits are dH/d$ and d 2 K/d$ 2 . We may therefore use (12*4) throughout for the real assembly; and at once obtain the following ex- pressions : ET c= g|e/5 =* P * T ' • (12 ' 5) _ _ p ^niogjiy * e _ i{iog 1/S) „ ( „ 2+o2+K2) dw dy dz du dv dw ^ (12-51) = P(^JV^<" 2+ ^>&....^. . . . (12-52) The temperature $ is determined by E = M4log/(3)+fP— L^, . (12-53) and the fluctuation of energy of the molecules in a bath of temperature S is given by (E -E c )?=a^E c =fPOT . . (12-54) If the fluctuation of c^ is evaluated, it takes the simple value (e t -Z t y = c ( , (12-55) whether it is in a bath or not ; for the second factor, analogous to that in (8*4), can be omitted when the cell is taken to be of small size. Thus in all cases the possession of c t is a normal property of the assembly. These results can be readily extended to cases where there is an external field of force acting on the molecules. By means of this assembly we can establish the meaning of 3 in terms of T, by observing that the gas itself constitutes a constant volume gas thermometer. It is easy to show that the pressure of a gas must be f of the mean kinetic energy the Partition of Energy, 477 in unit volume, that is to say, Jt> = P/ Vlog(l/$). Since the gas temperature is measured by the relation pV = P£T, we are again led to the relation d = e~ lik . We may observe that it is now possible to drop the assumption of commensurability, which was necessary in the sections which dealt with quantized systems. It was there essential, physically speaking, in order that it should be possible that the whole of the energy should be held somewhere; but as we now have molecules which can hold energy in any amounts, it may be dispensed with, the modifi- cation being justified on the same assumptions and by the same sort of limiting process as have been used in this section. Again, we can see that the correct results are obtained if H(~) replaces h(z) in (12*2) and all the other integrals, even though the interpretation as coefficient in a power series is no longer possible, and though the integrand is no longer single valued. In such many- valued integrands the limiting process shows that we simply require to take that value which is real on the positive side of the real axis. § 13. The Maxwell Distribution Law. We have carried out the whole process so far with quantized systems included in the assembly, but it may be observed that it is immediately applicable to an assembly composed solely of molecules. If this is done the value of c t in (12*51) establishes at once the Maxwell distribution law, and its- fluctuation in (12 # 55) proves that it is a normal property of the assembly. This is probably the simplest complete proof of the ordinary distribution law ; its special advantage is that by means of the fluctuations it is easily established that the actual distribution will hardly ever be far from the average. The method can also be made to establish the distribution law for a mixture of gases *, and indeed for a mixture of any kind, provided that the systems can be considered to have separate energies. It is also possible to extend the method to cases in which the total momentum or angular momentum is conserved, by constructing partition functions in more than one independent variable. In fact, there will be as many independent variables as there are uniform integrals of the dynamical equations of the assembly. For simplicity we shall suppose that the linear momentum in a given direction is conserved, * The effects of the semi-permeable membranes of thermodynamics- can be conveniently treated by the partition function. 478 On the Partition of Energy. and let its total amount be Gr. The method now requires the averaging process to be applied to expressions depending on P! W 2 , where we now have not only but also 2, t c t fi t =Q, where fi t is the momentum in the given direction of a mole- cule in the tth cell. To sum the appropriate expressions we must take as our partition function With this function C will be the coefficient of z a G in [h(z, #)] P , and this can be expressed as a double contour integral. So can the other averages, and the usual asymptotic expansions can be found. The correct distribution law follows on replacing h(z, x) by the integral which is its limit when the sizes of the cells tend to zero. This subject lies rather outside the theme of the present paper and need not be elaborated further. § 16. Summary. The whole paper is concerned with a method of calculating partitions of energy by replacing the usual calculation, which obtains the most probable state, and is mathematically un- satisfactory, by a calculation of the average state, which is the quantity that is actually required and which can be found with, rigour by the use of the multinomial theorem together with a certain theorem in complex variable theory. After a review of principles and two preliminary examples ihe real point of the method is illustrated in § 5. Here there are two groups of interacting Planck vibrators of different types. It is shown that the partition can be found by evaluating the coefficient of a certain power of z in an expression which is the product of power series in z. This coefficient can be expressed as a contour integral and can be evaluated by a well-known method, the " method of steepest descents." The result expresses itself naturally in terms of a parameter 3 which is identified with temperature measured on a scale given by ^~e~ llk . The work is extended to cover the partition among more The Heterodyne Beat Method. 479 general quantized systems in §9, and examples are given. In § 12 it is shown how it may be made to deal with as- semblies composed partly of: free molecules and partly of quantized systems. In §13 we deal with extensions possible when only molecules are present. The methods we have described can also be made to throw an interesting light on the statistical foundations of therrao- dynamics; but in that connexion many points have arisen which require rather careful discussion, and in order not to make the present paper too long, we have deferred them to a future communication. Cambridge, May, 1922. XLV. The Heterodyne Beat Method and some Applications to Physical Measurements. By Maurice H. Belz, M.Sc. (Cantab.), Barker Graduate Scholar of the University of Sydney *. IN a recent paper f, a preliminary account was given of the application of the heterodyne beat method to the measurement of magnetic susceptibilities. In virtue of the importance of the method as a sensitive measure of physical quantities, it seems desirable to give a more complete account of the principle and of some of the difficulties encountered in its application. Essentially the method consists of the following arrange- ment shown in fig. 1. Two oscillating circuits, Set 1 and Set 2, are set up side by side and arranged so as to have approximately the same frequency. The two sets are loosely coupled so that in the telephone included in one of the circuits a resultant beat frequency is maintained equal to the difference between the frequencies of the fundamentals or overtones in the two circuits. If symmetry in the two circuits is essential, direct coupling can be replaced by indirect coupling by means of a third circuit in which the telephone is placed. In either case, when the beat frequency is low enough, an audible note will be heard in the telephone, and any changes in the constants of either circuit will cause the frequency of the audible note to alter by an amount equal to the change in frequency of the responsible circuit. This at once provides * Communicated by Professor Sir E. Rutherford, F.R.S. t Belz, Proc. Camb. Phil. Soc. vol. xxi. part 2 (1922). 480 Mr. M. H. Belz on the Heterodyne Beat Method a very sensitive method. It is now easily possible to maintain oscillations of frequencies up to 10 7 per second. Taking the case when Set 1 has a frequency of 1,000,000 per second, Set 2 a frequency of 1,001,000 per second, the audible note will have a frequency of 1,000 per second. If the frequency of Set 1 is changed to 1,000,001 per second, the frequency of the audible note will now be 999 per second, and this change in pitch can readily be observed by comparison with a note of standard pitch. Fig. 1. This sensitive method has been successfully employed by Herweg *, Whiddington f , Pungs and Premier J, Falcken- burg §, and several others in physical researches, but the precautions necessary for steadiness in the beat note have never been completely specified. Precautions. With high frequency oscillations of the order 3 x 10 5 per second to 5 X 10 5 per second such as were used in the present investigations, electrostatic shielding from all external in- fluence was of the first importance. This was ensured by placing all the elements of the circuits in earthed metal- lined boxes, one of the variable capacities, by means of which final small adjustments were made, being provided with a long ebonite spindle which projected beyond the containing box. With the box closed the note from the telephone T was considerably reduced in intensity, and in order to obtain the maximum loudness, a small section was removed from the box, shielding being maintained by means of a piece of fine metal gauze. * Herweg, Zeit.f. Phys. vol. iii. p. 36 (1920). f Whiddington, Phil. Mag. vol. xi. p. 634 (1920). % Pungs and Preuner, Phys. Zeit. vol. xx. p. 543 (1919). § Falckenburg, Ann. d. Phys. vol. lxi. 2, p. 167 (1920). and some Applications to Physical Measurements. 481 Solidity of foundation is a most important requirement. In the experiments of Whiddington*, although the apparatus was set up on a solid base, vibrations of the building even at 2 a.m. proved troublesome. A somewhat similar trouble was experienced in some of the earlier experiments when the apparatus was installed on the top floor of the laboratory. It was found that the vibration of the building caused by people walking about the corridors, and by the passage of heavy motor traffic, appreciably affected the steadiness of the note. Although some of the work was done during the night and over the week end, the trouble always persisted. Finally the apparatus was transferred to a room on the ground floor and supported on stone pillars by means of solid rubber pads. The trouble was now completely removed so that successful observations could be made during the day despite the fact that people were continually walking beside the apparatus. After lighting the valves, a certain amount of time must elapse before the oscillating system has settled down to a steady state. This initial variation is due to the heating and expansion of the elements of the valves, causing changes in the whole capacity linked with the oscillating systems. In order to save time, thus sparing the high tension batteries and prolonging the life of the valves, the latter were contained in small tin boxes, placed outside the large box, which were lagged with asbestos and cotton -wool. In this way the heat conduction was minimized, and the valves settled down much more rapidly. Other conditions being the same, it was found that certain valves were less satisfactory than others. For some types the settling down process was very long, and by the time the valve was set, other things began to vary. After long trials with " E," " Fotos," and " A. T. " types, it was found that " B " type valves manufactured by the General Electric Company gave most satisfactory results, settling down most rapidly and remaining steadiest. The effects of the changes in the elements of an oscillating system on the frequency have been examined by Eccles and Vincent f in the case of wave-lengths of 3000 metres. They determined that between certain limits for each value of the coupling between the plate and grid coils there was a particular value of the filament current for which the wave- length was a maximum. Working at this value of the current it was found possible to hold the beat note steady to * Whiddington, loc. cit. | Eccles and Vincent, Proc. Roy. Soc. A. vol. xcvi. p. 455 (1919). Phil. Mag. S. 6. Vol. 44. No. 261. Sept. 1922. 2 I 482 Mr. M. H. Belz on the Heterodyne Beat Method one part in 100,000 for several minutes in spite of small unavoidable variations. With the frequencies employed in the present experiments, however, such a condition could not be established. The heating of the valve parts and the consequent change in capacity in the system resulting from changes in the filament current cause changes in wave-length which certainly far outweigh any real change due to increased thermionic emission alone. In order then to secure a constant filament current, accumulators of 100 ampere-hours capacity were employed. These were charged regularly after about three days' use, and after the valve had settled down, the current from them showed no variation during a single run. Faulty contacts of wires joining the elements of the circuits were avoided by soldering, the only sources of uncertainty being the sliding resistances in the filament circuits. These, however, were good types with bright surfaces and stiff springs so that the chance of error due to change of contact was small. The principal cause of variation in the frequencies of the circuits was found to be due to variations in the high tension batteries. This trouble has been mentioned by Eccles and Vincent *. In the present work the plate voltage was obtained from trays of portable accumulators of fairly low capacity, each tray providing 40 volts. After the valves had been burning for an hour or so, taking a current of about 10 milliamperes, this voltage began to vary and the beat note consequently drifted. However, giving the valves time to settle down, a matter of 15 to 20 minutes, it was found possible to hold the heterodyne note quite steady for intervals of 30 to 60 seconds, and this is ample time in which to make a single observation. After about 90 minutes burning the variation was too rapid and the batteries had to be recharged. The size and consequent capacity of the cells of these batteries is limited by the fact that they have to be contained in a metal box, and thus this source of variation can only be provided for in special cases. Technique. It is essential to maintain the oscillations generated in the circuits at frequencies considerably different from the natural frequencies of the coils alone f, that is to say with an appreciable capacity in the system, and under these conditions the frequency, n, of the oscillations in such a circuit containing * Eccles and Vincent, loc. cit. + Cf. Townsend, Phil. Mag. vol. xlii. August (1921). and some Applications to Physical Measurements. 483 inductance L and capacity C, is given very approximately by 71 = 1/(2^01). Changes in n can thus be brought about by changes in ( ! or L. In the experiments to be described below, the changes in n were brought about by variations in L, and in this case, with C constant, a small variation, dh, in the inductance produces a corresponding change, tin, in the frequency given bv dn/n=--idL/L (i.) The experimental part thus reduces itself to a determination of dn. This is accomplished by obtaining beats between the heterodyne note and a note o£ constant pitch, and then counting the change in the number of beats per second caused by the change in inductance. A considerable amount of practice in listening is required in order readily to be able to adjust the heterodyne note to the pitch of the constant note. This note can be very conveniently obtained by means of a third set, some distance away from the other sets, oscillating with audible frequency, in the plate circuit of which a telephone is placed. The intensity of the note heard can be altered by adjusting the filament current, and in this respect the note is very much more convenient than that obtained from a tuning-fork. For it was found that the heterodyne note could be more easily brought to tune with the standard note, and false beats more readily recognized when this latter could be altered so that both notes had approximately the same intensity. In some of the experiments * it was found impossible to obtain a beat note of convenient audible frequency when the fundamental frequencies of the oscillations were approxi- mately the same. It was observed that, as the capacity of one of the sets was altered, only very shrill notes could he heard on either side of the very large region of silence. This synchronization effect appears to depend on several factors, but chiefly on the coupling between the circuits. On account of the limited size of the box containing the coils, the coupling could not be reduced beyond a certain lower limit, and reducing the strengths of the oscillatory currents merely reduced the intensity of the limited note. In all these cases it was possible to obtain the heterodyne note between the fundamental of Set 2 and the first overtone of Set 1, which was quite steady and possessed the normal region of silence. Since the changes in n were produced by * Those in which the determinations of the magnetic susceptibilities of certain salts were made, see below. 2 I 2 484 Mi\ M. H. Belz on the Heterodyne Beat Method variations in the inductance o£ Set 1, this arrangement increases the sensitiveness of the method. For let "N" be the frequency, determined at the centre of the region of silence, of the fundamental oscillation in Set 2, n the frequency of the fundamental oscillation in Set 1, then since the rirst overtone of Set 1 is employed to produce the note, ~N = 2n« Let the frequency of the audible note from the third circuit be m. Then when the heterodyne note is adjusted, by slightly varying the capacity of Set 2, so that q beats per second are counted, the frequency of Set 2 is (N±??2±g) — (2n + m±q). If now the frequency of the fundamental oscillation of Set 1 is altered by dn per second, the frequency of the first overtone is altered by an amount 2dn per second, so that the frequency of the heterodyne note is now (2n±m + ^)~ (2n±2dn) — (m + q±2dn), whence if a change of p beats per second is observed when the induc- tance change is accomplished, p = 2dn. The sensitiveness is thus doubled, and could similarly be increased by em- ploying higher overtones of Set 1. Against this, however, is the fact that the notes so obtained are very feeble, and counting becomes increasingly difficult. From equation (i.) we see that the sensitiveness depends on n. It is now possible to maintain oscillations of frequencies up to 10 7 per second, but in cases where the change in in- ductance is caused by inserting a specimen within the coil, there is an upper limit to n determined by the form and function of the coil L. It is necessary to divide this coil into two parts between which there is no mutual inductance, one part L t being coupled to the grid circuit in order to maintain the oscillations, the other part L 2 serving as the coil in which the inductance changes occur. This latter part must be a fairly long coil in order that there may be an appreciable region within it through which the magnetic field is constant, in which region the specimen is placed. On account of the dimensions of this coil, the first part has to possess a fairly large inductance in order to get sufficient mutual inductance with the grid coil : further, a certain amount of coupling is required with Set 2 to produce the heterodyne note. Experimental. In the present experimental arrangements the details of the coils are as follows :— Coil Lj. The coil was 10 cm. long, and consisted of 100 turns of 22 s.w.g., double cotton covered. It was and some Applications to Physical Measurements. 485 wound on a short length of glass tubing and had an effective diameter of 2'10 cm. The self-inductance, employing the exact formula of Nagaoka *, viz., L s = 47r 2 <2 2 >i 1 2 b. K, where L s is the self-inductance of a current sheet of the same dimensions as the coil, n 1 the number of turns per cm., a the effective radius, b the total length, and K a factor depending on the ratio of the diameter of the coil to the length, to which was applied the correction for spacing, was calculated to be 39,160 cm. The small frequency correction was neglected. Coil L 2 . The coil was 36*70 cm. long, and consisted of 541 turns of copper wire, no. 24 s.w.g., silk covered. It was wound on a long glass tube, of external diameter 1*00 cm., and separated therefrom by means of a layer of paraffined paper. The effective diameter (2a) of the coil was 1*105 cm., and self-inductance, calculated as above, was 92,430 cm. The total inductance L ( = L 1 + L 2 ) is thus 131,600 cm. The coil L 2 was outside the box containing the rest of the circuits, and was shielded from external electrostatic influences by means of an enveloping earthed metal cylinder. Coil L 3 . The length was 9 cm., and the coil consisted of 90 turns of copper wire. no. 22 s.w.g., double cotton covered. It was wound on a short, length of glass tubing and had an effective diameter 4*13 cm. The self-inductance was similarly cal- culated to be 124,600 cm. The capacities employed had a range of 100 to 1200 microfarads and were provided with a slow movement. Changes in the frequency of the oscillations of Set 1 brought about by the insertion of a specimen within the coil L 2 may be due to three causes : — (a) In the first place, if the coil is not shielded from the electrostatic effect of the specimen, the self capacity of the coil will be changed. In order to observe changes of inductance alone, it is necessary to guard against this possibility. This was done by depositing a thin layer of platinum f on the outside of the glass tube on which the coil L was wound, and earthing. The thickness of the deposit, obtained by weighing, was 7 x 10 ~ 6 cm. It is necessary to * Nagaoka, Jour. Coll. Sci. Tokyo, xxvii. art. 6, p. 18 (1909). t The function of the paraffined paper was to prevent any possible short-circuiting 1 of the coil through the layer of platinum. 486 Mr. M. H. Belz on the Heterodyne Beat Method determine the effect of this shield on the strength of the magnetic field within. The magnetic force, H t , at a depth t in a mass of metal is related to the force, H , at the surface by the equation H, = IV V -ST-' in which //,, cr represent the permeability and specific resistance respectively of the metal, and p = 27rn, n being the frequency. Taking w = 4'84 x 10° per second, the largest frequency used, and